Associated Topics || Dr. Math Home || Search Dr. Math

### Geometric Proof of the Steiner-Lehmus Theorem

```Date: 05/13/2002 at 01:42:14
From: Jim
Subject: Plane Geometry

Prove:  If two bisectors of two angles of a triangle are equal then
the triangle is an isosceles triangle.

What I am looking for is a "geometric proof" where algebra isn't
involved.  I suppose if one were to use trigonometry and analytic
geometry the proof would be even easier.

I've seen a geometric proof that was quite long and involved but I've
misplaced it.  But it seems as if such a simple problem should have a
simple (geometric) solution!

```

```Date: 05/13/2002 at 16:16:52
From: Doctor Floor
Subject: Re: Plane Geometry

This is called the Steiner-Lehmus theorem.  Here is a geometric proof:

We consider given that ABC is a triangle, AD and BE bisect angles A
and B such that AD = BE.

Start by constructing F such that AF = AE and DF = AB. Let G be the

We find ADF and EBA are congruent, so

In triangles AGE and BGD we find

<AEB + 1/2 <A = <ADB + 1/2 <B

<FAB = <BDF

We also find that these final angles are

<AEB + 1/2 <A = <AGB

= 180 -(<A + <B)/2 > pi/2

From this we can conclude the congruence of BAF and FDB despite only
having SSA, because the unknown angles must be acute. So DB = AF = AE
and thus 1/2 <A = 1/2 <B, and we see that ABC is isosceles.

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/

```

```Date: 05/14/2002 at 02:08:55
From: Jim
Subject: Plane Geometry

Thank you very much!  That was exactly what I was looking for.  Very
short and not too complicated.  Bravo.

```

```Date: 10/14/2003 at 00:34:22
From: Daniel
Subject: Geometric Proof of the Steiner-Lehmus Theorem

I was happy to find a simple geometric proof of the Steiner-Lehmus
theorem on your site, but as I was reading through the proof there
was one step which I just could not understand how it was done.

In triangles AGE and BGD we find

<AEB + 1/2 <A = <ADB + 1/2 <B

<FAB = <BDF    <----- This is the step

I do not understand how this follows from the previous steps.  It

Any help would be appreciated,

Daniel

```

```Date: 10/14/2003 at 12:04:09
From: Doctor Peterson
Subject: Re: Geometric Proof of the Steiner-Lehmus Theorem

Hi, Daniel.

The proof should state which of two possible points is to be chosen
as F; when I tried going through the theorem just now, I made the
F' below):

C
/ \
/   \
/     \
/       \
/         \
/           \
/             \
/               \
F-----------------                \
\             E  -----------------D
\            /   \             / / \
\          /       \       /   /   \
\         /            G      /     \
\       /          /     \  /       \
\     /        /          /\        \
\    /      /            /    \      \
\  /   /               /         \   \
\ / /                  /            \ \
A--------------------/----------------B
\                 /
\             /
\          /
\      /
\  /
F'

With F on the same side of AD as C, as shown, the angles add up
so that

<FAD + 1/2 <A = <FAB

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/

```
Associated Topics:
High School Triangles and Other Polygons

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search