Locus of a point.

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Locus of a point.

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Source: Stars of Mathematics 2012, Seniors, Problem 2

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#1 h Mar 9, 2015, 7:20 PM • 3 Y

Y by FlakeLCR, Adventure10, Mango247

Let $\ell$ be a line in the plane, and a point $A \not \in \ell$ . Also let $\alpha \in (0, \pi/2)$ be fixed. Determine the locus of the points $Q$ in the plane, for which there exists a point $P\in \ell$ such that $AQ=PQ$ and $\angle PAQ = \alpha$ .

(Dan Schwarz)

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#2 h Mar 9, 2015, 10:35 PM • 2 Y

Y by Adventure10, Mango247

Let $\mathbf{R}(O, \theta)$ be the rotation with center $O$ and angle $\theta$ .
Let $\mathbf{H}(O, t)$ be the homothety with center $O$ and coefficient $t$ .

Then the locus of $Q$ is the image of $\ell$ under $\mathbf{R}(A, \alpha) \circ \mathbf{H}(A, \frac{1}{2\cos \alpha} )$ and $\mathbf{R}(A, -\alpha) \circ \mathbf{H}(A, \frac{1}{2\cos \alpha} )$ .

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