T is an equilateral triangle with side length equal to the answer to last week’s Possible Parabolas problem. P is a point in the interior of T such that the distance from P to the farthest vertex of T is twice the distance from P to the nearest vertex of T, and the distance from P to the third vertex of T is 3/2 the distance to the nearest vertex. Let Q, R, and S be the feet of the perpendiculars from P to each of the three sides of T.
What is the sum of the distances PQ + PR + PS?
Isosceles triangle ABC, with side BC half the length of each of the other two sides, is inscribed in circle Q. Ray r from point A is drawn so that side AC bisects the angle between BA and r. Ray r intersects circle Q again at point D.
What is the ratio of the length BD to BC?