Finding the largest possible value of an angle

Hi all,

I received this geometry question as a challenge several months ago and decided to pull it back out. I managed to prove that the angle is less than 45 degrees, but I have no idea what do do overall to find the answer.

ADOC is a secant of a circle center O. A lies outside the circle and D and C lie on the circle. ABE is also a secant of Circle O and B and E are distinct points lying on the circle O. If AB = BC, compute, in degrees, the largest possible possible integer value of the measure of angle <CAE.

Re: Finding the largest possible value of an angle

The order of the letters is important. ADOC implies that A is on an extension of a diameter of the circle, (on the D side), and ABE implies that the point B lies between A and E.

The angle CAE is equal to the angle ACB, so choose to maximise this angle instead.

Start by picturing the construction for a small value of ACB and argue that the angle can be increased upto the limiting case when B and E come together and A(BE) is a tangent to the circle at BE.

(Beyond that, either B and E switch places or B moves back towards D, whichever way you choose to look at it.)

From there it's relatively easy to calculate the angle. It turns out to be thirty degrees.

Re: Finding the largest possible value of an angle

The answer is 29 degrees, so you are close but you are close but not correct. I have no idea how one gets to 29 degrees though.

Re: Finding the largest possible value of an angle

It's to do with the statement that B and E are distinct points.

That means that the final limit of B tending to E can't be reached. In that case the answer would be 29.99.... degrees.

The final part of the last sentence then asks for the largest integer value, so that would be 29.

If we are allowed to proceed to the limiting point, the answer is 30.

Re: Finding the largest possible value of an angle

Ah, that makes sense. Thanks.