(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A circle is inscribed in a triangle

Here is a picture

Picture of circle inscribed in triangle, not necessarily to scale

Which is larger: the circumference of the circle, or the perimeter of the triangle?

2. Relevant equations

C=∏D (D=diameter of the circle, C=circumference of circle)

P=AB+BC+CA (P stands for perimeter, refer to the linked photo to see A,B, and C)

3. The attempt at a solution

This is taken out of Nova's Math GRE Bible. Their solution is as follows, but I have a dispute with the solution:

"From the figure, it is clear that to go from one point on the circle, say, point P to another point, say, point

Q, the shortest available path is the arc PQ. Hence, arc PQ < PA + AQ. Similarly, arc QR < QB + BR, and

arc RP < RC + CP. Summing the three inequalities yields arc PQ + arc QR + arc RP < (PA + AQ) +

(QB + BR) + (RC + CP). The right side of the inequality is the perimeter of the triangle ABC, and the left side is the circumference of the circle. Hence,

Column A is greater than Column B, and the answer is [the perimeter of the triangle is larger]."

What I'm disputing is their statement: "it is clear that to go from one point of the circle P to Q the shortest available path is the arc PQ" Wouldn't the shortest path be the chord PQ (because the shortest path between two points is a straight line, which, in that case would be the chord PQ NOT the arc)? I know that, for example the chord PQ < PA + AQ because we can form a triangle APQ and we know each side has to be less than the sum of the lengths of the other side. We can continue forming these small triangles and then end up with, just as they were doing

PQ + QR + RP < PA + AQ + QB + BR + RC + CP

where the right side is the perimeter of the big triangle, but now the left side is only the perimeter of a smaller triangle, PQR, that is inscribed in the circle!

Where do you get the relationship that the arc(PQ) < PQ + AQ as they are stating in this proof?

Thank you! I am very confused by their proof

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# Homework Help: Proof: circumference of circle inscribed in a triangle < perimeter of the triangle

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