Proofs In Plane Geometry
Properties Of Circles
1. tan.
rad.
2. rt.
in semicircle
3.
s in same seg.
4.
at centre = 2
at circ.
5.
s in opp. seg. (
)
6. tan. from ext. pt.
![Rendered by QuickLaTeX.com \perp](https://teach.sg/wp-content/ql-cache/quicklatex.com-d766824b5213bf1b1ae5f8209a631019_l3.png)
2. rt.
![Rendered by QuickLaTeX.com \angle](https://teach.sg/wp-content/ql-cache/quicklatex.com-16cc64d9eb5f830b5afc042e8f314306_l3.png)
3.
![Rendered by QuickLaTeX.com \angle](https://teach.sg/wp-content/ql-cache/quicklatex.com-16cc64d9eb5f830b5afc042e8f314306_l3.png)
4.
![Rendered by QuickLaTeX.com \angle](https://teach.sg/wp-content/ql-cache/quicklatex.com-16cc64d9eb5f830b5afc042e8f314306_l3.png)
![Rendered by QuickLaTeX.com \angle](https://teach.sg/wp-content/ql-cache/quicklatex.com-16cc64d9eb5f830b5afc042e8f314306_l3.png)
5.
![Rendered by QuickLaTeX.com \angle](https://teach.sg/wp-content/ql-cache/quicklatex.com-16cc64d9eb5f830b5afc042e8f314306_l3.png)
![Rendered by QuickLaTeX.com a+c=b+d=180^\circ](https://teach.sg/wp-content/ql-cache/quicklatex.com-bf7275548150b241d71114f545b78f1a_l3.png)
6. tan. from ext. pt.
Congruent & Similar Triangles
Midpoint Theorem
If
and
are the midpoints of
and
respectively, then
//
and
.
![Rendered by QuickLaTeX.com D](https://teach.sg/wp-content/ql-cache/quicklatex.com-c10ec9debc8ec5dce4c3c5887557202d_l3.png)
![Rendered by QuickLaTeX.com E](https://teach.sg/wp-content/ql-cache/quicklatex.com-638a7387bd72763290cc777a9b509c38_l3.png)
![Rendered by QuickLaTeX.com AB](https://teach.sg/wp-content/ql-cache/quicklatex.com-655cfe80f708bdc3fc9c95735c974c1f_l3.png)
![Rendered by QuickLaTeX.com AC](https://teach.sg/wp-content/ql-cache/quicklatex.com-384f3b401d6739513acd472e38839920_l3.png)
![Rendered by QuickLaTeX.com DE](https://teach.sg/wp-content/ql-cache/quicklatex.com-ae68afb045b08b1e4439aece66006bb7_l3.png)
![Rendered by QuickLaTeX.com BC](https://teach.sg/wp-content/ql-cache/quicklatex.com-49ab1bf3aae666ba4ce84548a3597eb4_l3.png)
![Rendered by QuickLaTeX.com DE=\dfrac{1}{2}BC](https://teach.sg/wp-content/ql-cache/quicklatex.com-2123ae5ec28b9442f6c4e6ee8889b510_l3.png)
Tangent-Chord Theorem (Alternate Segment Theorem)
If
is a tangent to the circle at
, then
and
.
![Rendered by QuickLaTeX.com DE](https://teach.sg/wp-content/ql-cache/quicklatex.com-ae68afb045b08b1e4439aece66006bb7_l3.png)
![Rendered by QuickLaTeX.com B](https://teach.sg/wp-content/ql-cache/quicklatex.com-c74288aabc0e2ca280d25d92bf1a1ec2_l3.png)
![Rendered by QuickLaTeX.com \angle CAB=\angle CBE](https://teach.sg/wp-content/ql-cache/quicklatex.com-8ab2600d29de722a7b62184294101ea6_l3.png)
![Rendered by QuickLaTeX.com \angle ACB=\angle ABD](https://teach.sg/wp-content/ql-cache/quicklatex.com-b5210cf5465de545c1ec302fab477e1a_l3.png)