Geometric Proofs
# ← Specialist Home
# Important Links
# Circle Proofs
# Arcs/Segment Proofs
- The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc
- Angles in the same segment of a circle are equal
- Angles in a semicircle are always $90^\circ$
- $\overline{AB}$ is the diameter
- $\overline{AB}$ is the diameter
- The angle at the centre of a circle is twice the angle at the circumference subtended by the same arc
# Tangent Proofs
- A line tangent to a circle is perpendicular to the radius drawn from the point of contact
- The two tangents drawn from an external point are the same length
- i.e $PT = PT’$
- i.e $PT = PT’$
- The angle between a tangent and a chord drawn from the point of contact is equal to any angle in the alternate segment
- Points $A$ and $B$ are stationary, but the point where they meet on the alternate segment can move and $\theta$ will always equal the angle between the tangent and the chord $\overline{AB}$
- A line tangent to a circle is perpendicular to the radius drawn from the point of contact
# Chord/Secant Proofs
- If $AB$ and $CD$ are two chords of a circle that cut at a point $P$, then:
- $PA \cdot PB = PC \cdot PD$
- This also works if $P$ is outside the circle
- $PA \cdot PB = PC \cdot PD$
- If $P$ is a point outside a circle and $T$, $A$, $B$ are points on the circle such that $PT$ is a tangent and $PAB$ is a secant, then $|PT|^2 = |PA| \cdot |PB|$
- If $AB$ and $CD$ are two chords of a circle that cut at a point $P$, then:
# Cyclic Quadrilateral Proofs
- A quadrilateral is cyclic if and only if the sum of each pair of opposite angles is 180$\degree$
- $a + c = 180^{\circ},\\ \\ b + d = 180^{\circ}$
- $a + c = 180^{\circ},\\ \\ b + d = 180^{\circ}$
- A quadrilateral is cyclic if and only if the sum of each pair of opposite angles is 180$\degree$