AMC 10 · 2021 · #24
Grade 11 geometry-2dPick an answer.
F sits at the end of a long chain — bisector, then one circle, then another circle — and chasing its position through that chain is punishing. Tool #16 (Change Focus) is the whole solution: stop asking where D, E, F are and ask only about products of distances measured from A. Both circles happen to speak that one language. The circumcircle of △ ABC says the bisector satisfies AD · AE = AB · AC; the circle through B, D, E, F says the two lines out of A cut it with the same product, AF · AB = AD · AE. Chain those and the unwanted quantities AD and AE cancel out without ever being computed. Tool #11 (Work Backwards) decides in advance where to aim: CF is a side of △ AFC, so the Law of Cosines will finish the job once AF and ∠ FAC are known — that is what makes AF the target worth hunting. Tool #1 (Draw a Diagram) is needed to see that F falls beyond B, which is what makes ∠ FAC equal to the triangle's own angle ∠ BAC. Tool #7 (Identify Subproblems) handles the two arithmetic pieces, each a single Law of Cosines: first cos∠ BAC from the three given sides, then CF itself.
Draw both circles and find F
Draw both circles and find the new point.
Drawing the second circle shows that F, for all the machinery used to build it, still lands on a line through A — so one number, AF, pins it down completely.
10.G-C.A.3Draw A DiagramWork backwards to triangle AFC
Work backwards to the target triangle.
Choosing the finishing triangle first turns a sprawling picture into a list of exactly two numbers worth chasing.
10.G-CO.A.1Work BackwardsGet cos A from the three sides
Get the cosine from the three sides.
Three sides determine a triangle completely, so any angle in it is one Law of Cosines away from the given numbers.
11.G-SRT.D.10Identify SubproblemsFirst circle: the bisector product AD·AE
The first circle gives the key product.
Equal inscribed angles on the same arc are what turn a circumcircle into a supplier of similar triangles, and similar triangles turn a ratio into a product.
10.G-C.A.2Change Focus Count The ComplementSecond circle: convert the product into AF
The second circle converts it into a length.
A point outside a circle sees every line through it with the same product of the two distances, so a product learned on one line hands you a length on another.
A point outside a circle sees every line through it with the same product of the two distances.
▸ Why?
The lines cut out triangles with matching angles, so their sides sit in one fixed ratio.
▸ Why?
Every point of the circle is one radius from the centre, which is what ties all the lines to one number.
Finish in the isosceles triangle
Finishing in the isosceles triangle gives 30.
Once the two sides out of the vertex turn out to be equal, the answer depends on nothing but that one length and the angle between them.
11.G-SRT.D.10Identify SubproblemsWhen a point is built by a chain of circles, do not chase where it is — chase the product of distances that every line through the outside point must share, because that product hands you the one length you actually need.
- Draw both circles and find F
- Work backwards to triangle AFC
- Get cos A from the three sides
- First circle: the bisector product AD·AE
- Second circle: convert the product into AF
- Finish in the isosceles triangle
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