Problem
GEO-B2-M01-P024 Two Parallel Tangents in Small Circles
In triangle \(ABC\), point \(D\) lies on side \(BC\). The tangent to circle \((ABD)\) at \(D\) is parallel to \(AC\). The tangent to circle \((ACD)\) at \(D\) is parallel to \(AB\). Prove that triangle \(ABC\) is equilateral.
The first tangent gives \(\angle A=\angle C\), and the second gives \(\angle A=\angle B\).
For circle \((ABD)\), the angle between the tangent at \(D\) and chord \(DB\) equals \(\angle DAB=\angle BAC\). But this tangent is parallel to \(AC\), and \(DB\) lies on \(BC\), so the same angle equals \(\angle ACB\). Hence \(\angle A=\angle C\).
For circle \((ACD)\), the angle between the tangent at \(D\) and chord \(DC\) equals \(\angle DAC=\angle BAC\). This tangent is parallel to \(AB\), and \(DC\) lies on \(BC\), so the same angle equals \(\angle ABC\). Hence \(\angle A=\angle B\). Thus \(\angle A=\angle B=\angle C\), and triangle \(ABC\) is equilateral.
The final problem of the module: two independent applications of the tangent-chord theorem in smaller circles.