Math Is Fun Forum

1. Two lines \ell and m intersect at O at an angle of 28^\circ. Let A be a point inside the acute angle formed by \ell and m. Let B and C be the reflections of A in lines \ell and m, respectively. Find the number of degrees in \angle BAC.

2. A laser is shot from vertex A of square ABCD of side length 1, towards point P on \overline{BC} so that BP = 3/4. The laser reflects off the sides of the square, until it hits another vertex, at which point it stops. What is the length of the path the laser takes?

Thanks!

Let the incircle of triangle ABC be tangent to sides BC, AC, and AB at D, E, and F, respectively. Prove that triangle DEF is acute.

I believe these are similar triangles...

1) In the figure with four circles below, let A_1 be the area of the smallest circle, let A_2 be the area of the region inside the second-smallest circle but outside the smallest circle, and so on. If A_1 : A_2 : A_3 : A_4 = 1 : 2 : 3 : 4, then find the ratio of the largest radius to the smallest radius.

2) In triangle ABC, AB = 5, AC = 6, and BC = 7. Circles are drawn with centers A, B, and C, so that any two circles are externally tangent. Find the sum of the areas of the circles.

3) Let ABCD be a square of side length 4. Let M be on side BC such that CM = 1, and let N be on side AD such that DN = 1. We draw the quarter-circle centered at A.

Let x and y denote the areas of the shaded regions, as shown. Find x - y.