The Varsity Math team members liked last week’s problem about a triangle so much that they clamored for more. Coach Newton is all too happy to oblige with two more triangular teasers.

### Equal Laterals

*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?*

### Isocircules

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*?

## Solution to week 36

**Circle Crossing**. Let *M* be the point of intersection of *P* and *Q* other than *F*; we shall show *M* also lies on *R*, establishing the answer as **0**. First note that angle *FME* subtends the arc of a full circle minus the measure of arc *FME*, which is just twice the measure of the angle at *A*. In other words, *FME* subtends an arc of 360 – 2*A* degrees, and so has measure 180 – *A* degrees. Similarly, angle *DMF* has measure 180 – *B* degrees. Subtracting these two angles from 360 degrees, we see that the measure of angle *EMD* is *A* + *B* degrees. But because the angles of triangle *ABC* sum to 180 degrees, the measure of angle *EMD* is therefore 180 – *C* degrees. But the arc of circle *R* from *E* to *D* clockwise is just the locus of all points which make an angle with *E* and *D* of 180 – *C* degrees, so point *M* lies on circle *R* as desired.

## Recent Weeks

**Week 36**: Possible Parabolas & Circle Crossing, solution to Heart Trick

**Week 35**: Hat Trick & Heart Trick, solutions to Test Test, Simultaneous Remainders, Different Dice, & No-Dispute Knights

**Week 34**: Different Dice & No-Dispute Knights, solution to Subtle Sequences

**Week 33**: Simultaneous Remainders & Subtle Sequences, solution to Late Show

**Week 32**: Test Test & Late Show, solutions to Talking Midpoints, Split Vote, Dream Sequence, & Alien Arithmetic

**Week 31**: Alien Arithmetic & Dream Sequence, solution to Treasure Trek

Links to all of the puzzles and solutions are on the Complete Varsity Math page.

**Come back next week** for answers and more puzzles.

[asciimathsf]