Problem
GEO-B1-M03-P014 Square of a Leg
#14
★★★☆☆ Level 3 of 5
In right triangle \(ABC\), angle \(C\) is right, and \(CH\) is the altitude to hypotenuse \(AB\). Prove that \(AC^2=AB\cdot AH\).
Prove that triangles \(ABC\) and \(ACH\) are similar.
Triangles \(ABC\) and \(ACH\) are right triangles: \(\angle C=90^\circ\), \(\angle AHC=90^\circ\). Also, the angle at \(A\) is common. Therefore \(\triangle ABC\sim\triangle ACH\). The side correspondence gives \(\frac{AB}{AC}=\frac{AC}{AH}\). Hence \(AC^2=AB\cdot AH\).
This is an important formula, but in this module it should be treated as a consequence of similarity.