Problem
GEO-B2-M01-P016 Two Circles on One Side
#16
★★★★☆ Level 4 of 5
Point \(D\) lies on side \(BC\) of triangle \(ABC\). Consider circles \((ABD)\) and \((ACD)\). Prove that the angle between these circles at \(D\) equals \(\angle BAC\) in oriented notation.
Draw the tangents to both circles at \(D\) and apply the tangent-chord theorem.
Let \(t_1\) be the tangent to \((ABD)\) at \(D\), and \(t_2\) the tangent to \((ACD)\) at \(D\). By the tangent-chord theorem, \(\angle(t_1,DB)\equiv\angle DAB\), and \(\angle(DC,t_2)\equiv\angle DAC\). Since \(DB\) and \(DC\) lie on one line, the angle between \(t_1\) and \(t_2\) equals \(\angle DAB+\angle DAC=\angle BAC\) in oriented notation.
A very useful configuration: two circles on parts of a triangle often give angle \(A\).