Steiner-Lehmus Theorem/Converse
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Converse to Steiner-Lehmus Theorem
Let $ABC$ be an isosceles triangle with $BA = BC$.
Let $D$ be the point of intersection of the angle bisectors through the vertices $A$ with the side $BC$.
Likewise, let $E$ be the point of intersection of the angle bisectors through the vertices $C$ with the side $BA$.
Then:
- $AD = CE$
Proof
We have:
| \(\ds BA\) | \(=\) | \(\ds BC\) | $ABC$ is isosceles | |||||||||||
| \(\ds \angle BCA\) | \(=\) | \(\ds \angle BAC\) | Isosceles Triangle has Two Equal Angles | |||||||||||
| \(\ds \angle DAC\) | \(=\) | \(\ds \frac 1 2 \angle BAC\) | $AD$ is an angle bisector | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 2 \angle BCA\) | Isosceles Triangle has Two Equal Angles | |||||||||||
| \(\ds \) | \(=\) | \(\ds \angle ECA\) | $CE$ is an angle bisector | |||||||||||
| \(\ds AC\) | \(=\) | \(\ds AC\) | Common Side | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \triangle ECA\) | \(\cong\) | \(\ds \triangle DAC\) | Triangle Angle-Side-Angle Congruence | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds AD\) | \(=\) | \(\ds CE\) | Definition of Congruence (Geometry) |
$\blacksquare$
Source of Name
This entry was named for Jakob Steiner and Daniel Christian Ludolph Lehmus.