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. AB = AC = BC = AP = BP = CP = 3 2 1 r + 1 r + 2 3 - r
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 Triangle PAC 2 1 r + 1 r + 2x
Law of Cosines
a2 = b2 + c2 – 2bcCos A
Triangle PAC

7. Now we will use law of cosines a second time2 1
r + 1
r + 2
3 - r x
Law of Cosines
Triangle PAC
Now we will use law of cosines a second time
This time we will use it on Triangle PAB

8. Law of Cosines - Triangle PAB2 1
r + 1
r + 2
3 - r x
Law of Cosines - Triangle PAB
a2 = b2 + c2 – 2bcCos A

9. 3 inches

10. Allows you to calculate the length of a cevian.
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.
Medians and altitudes are examples of “special” cevians in a triangle. The formula is:

11. We can rewrite the formula in a way that is easier to remember …
Stewart’s Theorem
A cevian is a line segment that extends from a vertex of a polygon to it’s opposite side.
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. Let’s plug our values in and see what we get
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
3 – r 2 1

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