Geometry · Right triangle · 30 60 90 triangle · Angle chasing · Equilateral triangle
In the right triangle \(ABC\) the right angle is at \(C\), the hypotenuse \(AB\) measures \(1\) dm, and \(\angle BAC = 30^\circ\). A point \(D\) inside the triangle satisfies
\[ \angle BDC = 90^\circ \qquad\text{and}\qquad \angle ACD = \angle DBA . \]
The line \(CD\) meets the side \(AB\) at \(E\). Find the length of the segment \(AE\).
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), izbirno (selection) round 2007, 1. letnik, category A, problem 3. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source