My math teacher gave these 2 bonus questions for extra credit. I have no idea where to even begin - could I get a few pointers to lead me in the right direction?
Each circle in an infinite sequence with decreasing radii is tangent externally to the one following it and to both sides of a given right angle. What is the ratio of the area of the first circle to the sum of the areas of the other circles in the sequence?
-Obviously the ratio of the radius of the first circle to the sum of the radii of the other circles is $1:\sqrt{2} -1$, but i am unsure of how to find the ratio for the areas.
Consider an isosceles triangle ABC with two sides equal to 1 and $ \angle A, \angle C=72^\circ $. Bisect $ \angle A $ with bisector AD and call its length $I_{1}$. Bisect $ \angle D$ with bisector DE and call its length $I_{2}$. Continue this process and determine $I_{1} + I_{2} + I_{3} + ...$
-All these triangles are similar, but I'm not sure of the ratio in which ABC is greater than ADC due to the fact we haven't learned trigonometry yet, so how should I go about solving this problem?