# Construct Triangle from Angle, Inradius, and Altitude

### Problem

Construct $\Delta ABC,$ given angle $A=\angle BAC,$ the inradius $r,$ and the altitude $h_a.$

### Analysis

The sideline $BC$ of $\Delta ABC$ serves as the common tangent to the incircle $(I)$ and circle $C(A,h_{a})$ centered at $A,$with $h_a$ as the radius.

### Construction

Start at the vertex $A$ with the given angle $A$ and then inscribe a circle $(I)$ of radius $r.$

Draw $C(A,h_{a})$ and find common external tangents for $(I)$ and $C(A,h_{a}).$ Choose one of the two. It intersects the sides of the angle $A$ in points $B$ and $C.$

The tangent may not exist if circle $(I)$ lies entirely within $C(A,h_{a}).$ If the given altitude is smaller than the inradius, the construction leads to a triangle for which $(I)$ is an excircle. The same happens if one chooses the inner instead of external common tangent of the two circles.

### Acknowledgment

The construction is due to Prof. Dr. René Sperb.