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Tangent Circles

Tangent Circles. Finding solutions with both Law of Cosines and Stewart’s Theorem. Inspired by (though not actually stolen from) problem #10 on the 2011 High School Purple Comet Competition. With thanks to Luke Shimanuki for sharing his solution. 3 inches.

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Tangent Circles

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  1. Tangent Circles Finding solutions with both Law of Cosines and Stewart’s Theorem Inspired by (though not actually stolen from) problem #10 on the 2011 High School Purple Comet Competition With thanks to Luke Shimanuki for sharing his solution

  2. 3 inches

  3. Find the following segments either as whole numbers or in terms of r (the radius of circle P) AB = AC = BC = AP = BP = CP = 3 2 1 r + 1 r + 2 3 - r

  4. AB = AC = BC = AP = BP = CP = 3 2 1 r + 1 r + 2 3 - r Let us now focus on two of the triangles – APC and APB I will redraw and enlarge the two triangles on the next slide

  5. r + 1 3 - r r + 2 x 2 1 We will begin by using Law of Cosines with triangle APC to solve for angle PAC (x degrees)

  6. Law of Cosines a2 = b2 + c2 – 2bcCos A r + 2 Triangle PAC r + 1 3 - r x 2 1

  7. Law of Cosines Triangle PAC r + 2 r + 1 3 - r x 2 1 Now we will use law of cosines a second time This time we will use it on Triangle PAB

  8. Law of Cosines - Triangle PAB a2 = b2 + c2 – 2bcCos A r + 2 r + 1 3 - r x 2 1

  9. 3 inches

  10. Stewart’s Theorem Allows you to calculate the length of a cevian. A cevian is a line segment that extends from a vertex of a polygon to it’s opposite side. a Medians and altitudes are examples of “special” cevians in a triangle. The formula is:

  11. Stewart’s Theorem A cevian is a line segment that extends from a vertex of a polygon to it’s opposite side. a We can rewrite the formula in a way that is easier to remember … A man and his dad put a bomb in the sink ….

  12. Stewart’s Theorem Let’s plug our values in and see what we get a = b = c = d = m = n = 3 r + 2 r + 1 a 3 – r 2 1

  13. Stewart’s Theorem a = b = c = d = m = n = 3 r + 2 r + 1 3 – r 2 1 a

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