Euclidean Geometry Basics

Geometry

Congruent Triangles

Two triangles are congruent if all sides and their corresponding angles are the same, denoted by

Congruence Conditions

Traveling around a triangle counter-clockwise, starting at angle, two triangle are congruent under the conditions the following match

However does not guarantee congruence unless the angle is . Similarly does not guarantee congruence as the side lengths can be scaled, but it is similar.

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Triangle Basics

Side Inequality

In a triangle, the sum of any two lengths is strictly greater than the length of the third.

Angle Inequality

Let be a triangle then .

Angles

The measure of an exterior angle is equal to the sum of the other tow interior angles.

Altitudes

An altitude is a line that passes through a vertex and is perpendicular to the opposite side (extended if necessary). The point at which the altitude intersects the opposite side is called the foot. All three altitudes are concurrent meaning the intersect at a single point called the orthocentre.

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Parallel Lines

Two parallel lines are denoted as . The line that intersects both lines is called a transversal. Alternate interior angles cut by a transversal are equal. The convers is true if two lines are cut by a third and the alternate interior angles are equal then the lines are parallel.

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Parallelograms

A Parallelogram is a quadrilateral whose opposite edges are parallel. The following properties follow:

1. Opposite edges have equal length.
2. Opposite angles are supplementary - sum to .
3. The diagonals bisect each other.

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Circles

Relationship Between Chords and Radii

Consider the following statements about a line, a circle and chord. If two of the following is true the third is.

1. The line passes through the centre of the circle.
2. The line passes through the midpoint of the chord.
3. The line is perpendicular to the chord.

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Circles and Triangles

Circumcircle

Given a triangle , the circumcircle or circumscribed circle is the circle that points lie on with centre which is called the circumcentre and is the circumradius. It is guaranteed to exist for all triangles and it’s centre is the intersection point of all three perpendicular bisectors of the sides of the triangle, making it unique.

Incircle

Given a triangle , the incircle is such that all sides of the triangle are tangent to it. The centre is called the incentre and the radius is called the inradius. It is guaranteed to exist for all triangles and it’s centre is the intersection point of all three angle bisectors, making it unique.