FullProof Help Center

The full list of theorems and reasons in FullProof ​

General

  • Given
  • Denote
  • Algebra / Calculation
  • Transitivity
  • Halves of equals are equal.
  • Angle equal to itself (reflexivity)
  • Segment equal to itself (reflexivity)
  • Area equal to itself (reflexivity)
  • Arc equal to itself (reflexivity)

Angles Segments and Lines

Angles and Segments

  • Angles around a point add up to 360°.
  • If two angles form a linear pair, then they are supplementary.
  • If two pairs of angles form two linear pairs, and the first angles in each pair are congruent, then the second angles in each pair are also congruent.
  • Adjacent angles on a straight line sum to 180°.
  • The legs of a straight angle form a straight line.
  • Vertical angles are congruent.
  • Angle addition/subtraction
  • Addition/subtraction of equal angles
  • Definition: angle bisector
  • A point on an angle bisector is equidistant from the sides of the angle.
  • A point equidistant from the sides of an angle lies on the angle bisector.
  • Line segment addition/subtraction
  • Addition/subtraction of equal line segments
  • Definition: midpoint
  • Definition: segment bisector
  • Definition: perpendicular bisector
  • A point on a perpendicular bisector is equidstant from the endpoints of the bisected segment.

Parallel Lines and Perpendicular Lines

  • Definition: perpendicular lines
  • Definition: parallel lines
  • The distance between two parallel lines is constant.
  • Segments of parallel lines are parallel.
  • Two lines parallel to a third line are parallel.
  • If two lines are perpendicular to a third line, then they are parallel to each other.
  • Perpendicular Transversal Theorem: A line perpendicular to one of the parallel lines, is perpendicular to both.
  • If two parallel lines are cut by a transversal, the same side interior angles are supplementary.
  • If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
  • If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
  • If two lines are cut by a transversal and the same side interior angles are supplementary, then the lines are parallel.
  • If two lines are cut by a transversal and a pair of corresponding angles is congruent, then the lines are parallel.
  • If two lines are cut by a transversal and a pair of alternate angles is congruent, then the lines are parallel.

Triangles

General Triangle Theorems

  • The sum of the interior angles of a triangle is 180°.
  • The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.
  • In a triangle, the longest side is across from the largest angle.
  • In a triangle, the largest angle is across from the longest side.
  • In a right triangle, the right angle is the largest.
  • In a right triangle, the hypotenuse is the longest side.
  • In an obtuse triangle, the obtuse angle is the largest.
  • Triangle Inequality Theorem
  • Definition: altitude
  • Definition: median

Congruence Theorems and Properties of Congruent Triangles

  • Side-angle-side (SAS) congruence
  • Angle-side-angle (ASA) congruence
  • Side-side-side (SSS) congruence
  • Side-side and an angle opposite to the largest side (SSA-largest side) congruence.
  • Angle-angle-side (AAS) congruence
  • Hypotenuse-leg congruence
  • Corresponding sides in congruent triangles are congruent.
  • Corresponding angles in congruent triangles are congruent.
  • In congruent triangles, the areas are equal.
  • In congruent triangles, the perimeters are equal.
  • In congruent triangles, the corresponding altitudes are congruent.
  • In congruent triangles, the corresponding medians are congruent.
  • In congruent triangles, corresponding angle bisectors are congruent.

Isosceles Triangle

  • Definition: isosceles triangle
  • Base angles in isosceles triangle are congruent.
  • A triangle with two congruent angles is isosceles.
  • If two sides of a triangle are congruent, then the angles opposite to these sides are congruent.
  • If two angles of a triangle are congruent, the sides opposite to these angles are congruent.
  • In an isosceles triangle, the vertex angle bisector, base median and base altitude coincide.
  • A triangle in which an altitude is a median is an isosceles triangle.
  • A triangle in which an altitude is an angle bisector is an isosceles triangle.
  • A triangle in which an angle bisector is a median is an isosceles triangle.

Equilateral Triangle

  • Definition: equilateral triangle
  • An equilateral triangle is equiangular.
  • An equiangular triangle is equilateral.
  • In an equilateral triangle, all angles equal 60°.
  • If all angles of a triangle equal 60°, then the triangle is equilateral.
  • An isosceles triangle with an angle of 60° is equilateral.

Right Triangle

  • Definition: right triangle
  • The Pythagorean Theorem
  • The Converse Pythagorean Theorem
  • In a right triangle, the median to the hypotenuse is equal to half the hypotenuse.
  • A triangle in which the median to a side is equal to half of that side is a right triangle.
  • In a 30-60-90 triangle, the short leg is half the hypotenuse length.
  • In a right triangle, if the short leg equals half of the hypotenuse length, then an angle opposite to that side is 30°.

Area and Circumference of Triangles

  • The area of a triangle equals half of the product of a length of a base and the corresponding height.
  • A median divides the area of the triangle in half.
  • If two triangles have a common height (or congruent heights), the ratio between their areas equals the ratio between their bases.
  • Two triangles on the same base (or congruent bases) and between the same parallels are equal in area.
  • Area addition/subtraction
  • Definition: perimeter

Quadrilaterals

General

  • Sum of angles in a quadrilateral equals 360°.

Kite

  • Definition: kite (a quadrilateral with two disjoint pairs of congruent adjacent sides)
  • The non-vertex angles of a kite are congruent.
  • In a kite, the main diagonal bisects a pair of vertex angles.
  • In a kite, the main diagonal bisects the other.
  • In a kite, diagonals are perpendicular.
  • In a kite, a main diagonal is a perpendicular bisector to the other diagonal.
  • A quadrilateral in which one diagonal is the perpendicular bisector of the other is a kite.

Parallelogram

  • Definition: parallelogram (a quadrilateral with two pairs of opposite parallel sides)
  • Opposite sides in a parallelogram are congruent.
  • Opposite angles in a parallelogram are congruent.
  • Adjacent angles of a parallelogram are supplementary.
  • Diagonals of a parallelogram bisect each other.
  • A quadrilateral in which both pairs of opposite sides are congruent is a parallelogram.
  • A quadrilateral in which both pairs of opposite angles are congruent is a parallelogram.
  • A quadrilateral in which every two adjacent angles are supplementary is a parallelogram.
  • A quadrilateral in which one pair of opposite sides is congruent and parallel, is a parallelogram.
  • A quadrilateral whose diagonals bisect each other is a parallelogram.
  • Definition: altitude

Rectangle

  • Definition: rectangle (a quadrilateral with four right angles)
  • A quadrilateral with three right angles is a rectangle.
  • An equiangular quadrilateral is a rectangle.
  • In a rectangle, opposite sides are parallel.
  • Opposite sides in a rectangle are congruent.
  • Diagonals of a rectangle are congruent and bisect each other.
  • Diagonals of a rectangle are congruent.
  • Diagonals of a rectangle bisect each other.
  • A parallelogram with a right angle is a rectangle.
  • A parallelogram with congruent diagonals is a rectangle.

Rhombus

  • Definition: rhombus (a quadrilateral with four congruent sides)
  • In a rhombus, opposite sides are parallel.
  • Opposite angles in a rhombus are congruent.
  • Adjacent angles of a rhombus are supplementary.
  • Diagonals of a rhombus bisect each other.
  • Diagonals of a rhombus are perpendicular to each other.
  • Diagonals of a rhombus bisect its angles.
  • A parallelogram with two congruent adjacent sides is a rhombus.
  • A parallelogram in which a diagonal bisects an angle is a rhombus.
  • A parallelogram with perpendicular diagonals is a rhombus.

Square

  • Definition: square (a quadrilateral with four congruent sides and four right angles)
  • In a square, opposite sides are parallel.
  • Diagonals of a square are congruent and bisect each other.
  • Diagonals of a square are congruent.
  • Diagonals of a square bisect each other.
  • Diagonals of a square bisect its angles.
  • Diagonals of a square are perpendicular to each other.
  • A rectangle in which two adjacent sides are congruent is a square.
  • A rectangle in which a diagonal bisects an angle is a square.
  • A rectangle with perpendicular diagonals is a square.
  • A rhombus with a right angle is a square.
  • A rhombus with congruent diagonals is a square.

Trapezoid

  • Definition: trapezoid (a quadrilateral a pair of parallel sides)
  • Definition: right trapezoid
  • Definition: isosceles trapezoid
  • In an isosceles trapezoid, base angles are equal.
  • The diagonals of an isosceles trapezoid are congruent.
  • A trapezoid with equal base angles is isosceles.
  • A trapezoid with equal diagonals is isosceles.
  • A trapezoid inscribed in a circle is isosceles.
  • Definition: altitude

Area and Circumference of Quadrilaterals

  • The area of a trapezoid equals the product of the height with half the sum of the bases.
  • The area of a parallelogram equals the product of the length of the base and the corresponding height.
  • The area of a rectangle equals the product of two adjacent sides.
  • The area of a rhombus equals the product of the length of the base and the corresponding height.
  • The area of a rhombus equals half the product of its diagonals.
  • The area of a square equals the square of its side.
  • The area of a square equals half the product of its diagonals.
  • The area of a kite equals half the product of its diagonals.
  • The perimeter of a parallelogram equals twice the sum of two adjacent side lengths.
  • The perimeter of a rectangle equals twice the sum of two adjacent side lengths.
  • The perimeter of a rhombus equals four times the side length.
  • The perimeter of a square equals four times the side length.
  • Area addition/subtraction
  • Definition: perimeter

Midsegments in a Triangle or a Trapezoid and Triangle Centers

Midsegment of Triangle

  • Definition: midsegment in triangle
  • Triangle Midsegment Theorem (A midsegment of a triangle is parallel to the third side and equals its half)
  • A segment through a midpoint of one side of a triangle and parallel to the second side, bisects the third side.
  • A segment hrough a midpoint of one side of a triangle and parallel to the second side, is a midsegment.
  • A segment connecting two sides of a triangle, which is parallel to the third side and equals its half, is a midsegment.

Midsegment of Trapezoid

  • Definition: midsegment in a trapezoid
  • The midsegment of a trapezoid is parallel to its bases.
  • The length of the midsegment of a trapezoid is half of the sum of the bases.
  • Trapezoid Midsegment Theorem (The midsegment of a trapezoid is parallel to the bases and its length is half their sum)
  • A segment through a midpoint of one leg of a trapezoid and parallel to its bases, is a midsegment of the trapezoid.
  • A segment through a midpoint of one leg of a trapezoid and parallel to its bases, bisects the second leg.
  • In a trapezoid, a segment which connects the legs, is parallel to the bases and is equal to half of their sum, is a midsegment.

Triangle Centers

  • Definition: point of concurrency
  • Definition: Centroid (The point of concurrency of the medians in a triangle)
  • The medians of a triangle are concurrent.
  • Centroid divides each median in a 1:2 ratio.
  • The three altitudes of a triangle are concurrent.
  • Definition: point of concurrency of angle bisectors
  • The angle bisectors of a triangle are concurrent.
  • Definition: point of concurrency of perpendicular bisectors
  • The three perpendicular bisectors of the sides of a triangle are concurrent.

Circles

General Circle Theorems

  • Three non-collinear points define a unique circle.
  • A point that is equidistant from three points on a circle is the center.
  • Definition: radius
  • Definition: diameter
  • All radii of a circle are congruent.
  • All diameters of a circle are congruent.
  • The diameter of a circle is twice its radius.
  • A center of a circle is a midpoint of any diameter.
  • The distance between a point inside a circle and its center is less than the radius.
  • The distance between a point outside a circle and its center is greater than the radius.
  • Arcs in a circle add up to 360°.
  • A diameter divides the circle into two equal arcs.
  • Definition: arc midpoint
  • Definition: arc bisector
  • Arc addition/subtraction

Central Angles Chords and Arcs

  • In a circle, congruent arcs have congruent central angles.
  • In a circle, congruent central angles have congruent arcs.
  • In a circle, congruent central angles have congruent chords.
  • In a circle, congruent chords have congruent central angles.
  • In a circle, congruent chords have congruent arcs.
  • In a circle, congruent arcs have congruent chords.
  • In a circle, the measure of an arc equals the measure of the corresponding central angle.

Perpendicular from Center of Circle to Chord and Chord Length

  • The line through the center of a circle, that is perpendicular to a chord bisects the chord.
  • In a circle, a radius perpendicular to a chord bisects the corresponding central angle.
  • In a circle, a radius perpendicular to a chord bisects the corresponding arc.
  • The segment connecting the centre of a circle and the mid-point of a chord is perpendicular to the chord.
  • In a circle, the perpendicular bisector of a chord passes through the center of the circle.
  • Chords that are equidistant from the center of a circle are congruent.
  • Congruent chords in a circle are equidistant from the center.
  • In a circle, the chords that are closer to the center of the circle are longer than the chords that are father from the center
  • If two chords of the circle are unequal, the longer chord is closer to the center of the circle than the shorter chord.

Inscribed Angle

  • In a circle, the measure of an inscribed angle is half the measure of the central angle intercepting the same arc.
  • In a circle, inscribed angles intercepting the same arc are congruent.
  • In a circle, inscribed angles intercepting the same chord on the same side of the chord, are congruent.
  • An angle inscribed in a semicircle is a right angle.
  • In a circle, an inscribed right angle intercepts a diameter.
  • In a circle, arcs corresponding to congruent inscribed angles are congruent.
  • In a circle, inscribed angles that intercept congruent arcs are congruent.
  • In a circle, chords subtending congruent inscribed angles are congruent.
  • In a circle, inscribed acute angles intercepting congruent chords are congruent.

Tangent to a Circle and Two Tangents to a Circle

  • A tangent to a circle is perpendicular to the radius drawn to the point of tangency.
  • A line perpendicular to a radius at a point on a circle is tangent to the circle.
  • A line perpendicular to a tangent at a point of tangency passes through the center of the circle.
  • If two tangents to a circle are parallel, the segment connecting the points of tangency is a diameter.
  • From a point exterior to a circle, the two tangent segments are congruent.
  • A segment connecting the center of a circle and the intersection of the two tangent lines bisects the angle between the tangents.
  • An angle between a chord and a tangent is congruent to an inscribed angle on the opposite side of the chord.

Circumscribed Circle and Inscribed Circle

  • Definition: circumcenter
  • Definition: incenter
  • Definition: inscribed circle
  • The point of concurrency of perpendicular bisectors of a triangle is a circumcenter of the triangle.
  • The point of concurrency of angle bisectors of a triangle is an incenter of the triangle.
  • In congruent triangles, the circumradii are congruent.
  • In congruent triangles, the inradii are congruent.
  • Definition: circumscribed quadrilateral
  • Definition: inscribed quadrilateral
  • In a cyclic quadrilateral, opposite angles are supplementary.
  • If opposite angles in a quadrilateral are supplementary, then it is cyclic.
  • A quadrilateral is circumscribable iff the sum of length of two opposite sides equals the sum of the length of the remaining two sides.

Two Circles

  • The segment between the centres of two intersecting circles is a perpendicular bisector of the common chord.
  • The point of tangency of two tangent circles lies on the line connecting their centers.

Similarity and Proportion

Thales Theorems

  • Parallel Projection Theorem
  • The Side-Splitting Theorem
  • The Converse of Parallel Projection Theorem
  • The Converse of the Side-Splitting Theorem

Angle Bisector Theorems

  • The Angle Bisector Theorem (The angle bisector in a triangle divides the side oppposite the angle in a ratio equal to the ratio of the sides)
  • The Converse Angle Bisector Theorem

Similar Triangles

  • Angle-angle (AA) similarity
  • Side-angle-side (SAS) similarity
  • Side-side-side (SSS) similarity
  • In similar triangles, the ratios of corresponding sides are equal.
  • Corresponding angles in similar triangles are congruent.
  • In similar triangles, the ratio of corresponding altitudes is equal to the ratio of corresponding sides.
  • In similar triangles, the ratio of corresponding medians is equal to the ratio of corresponding sides.
  • In similar triangles, the ratio of corresponding angle bisectors is equal to the ratio of corresponding sides.
  • In similar triangles, the ratio between the perimeters is equal to the ratio between corresponding sides.
  • In similar triangles, the ratio of the circumradii is equal to the ratio of corresponding sides.
  • In similar triangles, the ratio of the inradii is equal to the ratio of corresponding sides.
  • In similar triangles, the ratio of the areas is equal to the square of the ratio of corresponding sides.

Proportions in a Right Triangle

  • In a right triangle, the altitude to the hypotenuse froms two triangles that are similar to the given triangle and to each other.
  • In aright triangle, the length of an altitude to the hypotenuse is the geometric mean of the length of the two segments of the hypotenuse.
  • If the altitude is drawn to the hypotenuse of a right triangle, each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.

Proportions in a Circle

  • Intersecting Chords Theorem
  • Intersecting Secants Theorem
  • Tangent - Secant theorem

Trigonometry

These reasons appear in the list only in trigonometry questions.

  • Trigonometry in right triangle
  • In a right triangle, the sine of an angle is the ratio of the length of the opposite side divided by the length of the hypotenuse.
  • In a right triangle, the cosine of an angle is the ratio of the length of the side adjacent to the angle divided by the length of the hypotenuse of the triangle.
  • In a right triangle, the tangent of an angle is the ratio of the length of the opposite side divided by the side adjacent to the angle.
  • The area of a triangle equals half of the product of the length of two sides and the sine of the included angle.

Analytical Geometry

These reasons appear in the list only in analytic geometry questions.

  • Formula for distance between two points
  • Formula for the midpoint of a segment
  • If a line is parallel to the X axis, then all points on that line have the same Y coordinate.
  • If a line is parallel to the Y axis, then all points on that line have the same X coordinate.
  • If two points on a line have the same X coordinate, then the line is parallel to the Y axis.
  • If two points on a line have the same Y coordinate, then the line is parallel to the X axis.
  • If one line is parallel to the X axis, and another is parallel to the Y axis, then both lines are perpendicular to each other.
  • Parallel lines have the same slope.
  • Lines with equal slopes are parallel to each other.
  • The product of the slopes of two perpendicular lines equals -1.