Proof
The two sub-triangles each share an angle with the original (the
non-right angle at or ) and both have a right angle (at the
foot of the altitude and at the apex), so they are equiangular with
the original by I.32, hence similar by VI.4. By transitivity (V.11)
they are similar to each other.
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Depends on (3)
- I.32Proposition I.32In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles,…
- VI.4Proposition VI.4In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides which…
- V.11Proposition V.11Ratios which are the same with the same ratio are also the same with one another.
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