Proofs In Plane Geometry
Properties Of Circles
1. tan. $\perp$ rad.
2. rt. $\angle$ in semicircle
3. $\angle$s in same seg.
4. $\angle$ at centre = 2 $\angle$ at circ.
5. $\angle$s in opp. seg. ($a+c=b+d=180^\circ$)
6. tan. from ext. pt.
2. rt. $\angle$ in semicircle
3. $\angle$s in same seg.
4. $\angle$ at centre = 2 $\angle$ at circ.
5. $\angle$s in opp. seg. ($a+c=b+d=180^\circ$)
6. tan. from ext. pt.
Congruent & Similar Triangles
$$\begin{array}{|c|c|}
\hline
\text{Congruent triangles} & \text{Similar triangles}\\\hline
\text{SSS, SAS, AAS, RHS} & \text{SSS, SAS, AAA}\\\hline
\end{array}$$
\hline
\text{Congruent triangles} & \text{Similar triangles}\\\hline
\text{SSS, SAS, AAS, RHS} & \text{SSS, SAS, AAA}\\\hline
\end{array}$$
Midpoint Theorem
If $D$ and $E$ are the midpoints of $AB$ and $AC$ respectively, then $DE$ // $BC$ and $DE=\dfrac{1}{2}BC$.
Tangent-Chord Theorem (Alternate Segment Theorem)
If $DE$ is a tangent to the circle at $B$, then $\angle CAB=\angle CBE$ and $\angle ACB=\angle ABD$.
