April 8, 2016
1
Pearland MC
1
Preliminaries
1. You and a partner will take turns positioning a pencil underneath a sheet
of paper. The first person to balance the paper wins (if the paper bends
that’s acceptable).
|
→. For the rest of today, consider a triangle with vertices A, B, C. Construct
a circle containing A, B, C.
2. When does a circle contain AB? What are the possible centers?
3. When does a circle contain AB, AC? What are the possible centers?
4 (Circumcircle, circumcenter). Show that there exists a circle containing
A, B, C. This will be called the circumcircle of ABC, and its center the
circumcenter, often denoted as O.
|
→. Relate a circle to the three sides of a triangle. Is there always a circle
tangent to BC, CA, AB?
5. When is a circle tangent to AB? What are the possible centers?
6. When is a circle tangent to AB, AC? What are the possible centers?
What do they form?
7. When is a circle tangent to AB, AC, BC? How can you narrow down the
possible centers?
8 (Incircle, incenter). Show that there is a circle tangent to BC, CA, AB.
This will be called the incircle of ABC, and its center the incenter, often
denoted as I.
Recall the importance of perpendicular bisectors. Now, we’re going to
look at perpendiculars. Perpendiculars are actually quite important, not
because someone chose 90 as a random number between 0 and 180. Consider
what makes Euclidean geometry unique—parallels. Euclid (Ευκλείδης), in
the Elements, actually defined parallels using perpendiculars. Furthermore,
the coordinate plane is drawn with perpendicular axes. And it couldn’t be a
coincidence that perpendiculars came up while constructing possible centers.
Why is it so special? It sported symmetry.
April 8, 2016
Pearland MC
2
|
→. Construct D on BC where AD is perpendicular to BC. Similarly, construct E, F as the feet of the perpendiculars from B, C to CA, AB. AD,
BE, CF turn out to be concurrent (meet at one point).
9. Although AD, BE, CF are perpendiculars, what do AD, BE, CF have
to do with perpendicular bisectors? Construct a clever segment with AD as
its perpendicuar bisector.
10 (Orthocenter). Show that AD, BE, CF concur. This point will be called
the orthocenter of ABC, often denoted H.
Recall the importance of perpendicular bisectors. Now, we’re going to
look at the bisector of a segment, called its midpoint.
|
→. Construct U , V , W midpoints of BC, CA, AB. What do you notice
about AU , BV , CW ?
11. Where must AU intersect BV ? For example, can you describe this
intersection using AU alone?
12 (Centroid). Show that AU , BV , CW are concurrent. The point of conAG BG CG
, GV , GW .
currency is called the centroid G. Find GU
That’s a fundamental result in itself. Now, we’re going to take it a little
further, looking at these ratios.
|
→. Considering the ratios
ABC.
AG BG
CG
, GV , GW
,
GU
find a clever transformation on
13 (Euler line). Find three special points of ABC that lie on a line in a
certain ratio.
Exercises
14. Find another proof of the existence of the centroid. First, prove something about a special configuration of four points (AKA quadrilateral).
15. Find another proof of the existence of the orthocenter (AKA altitudes
are concurrent) using the centroid.
April 8, 2016
2
Pearland MC
3
Ratios
Ratios appear in similar triangles. To find some similar triangles, we have to
find some equal angles first. Which construction yields equal angles?
|
→. Let AX be the angle bisector between AB, AC, where X lies on BC.
?
Can you find BX
XC
16. Construct a triangle similar to ABX where the triangle contains XC
corresponding to XB. Find BX
.
XC
17. Suppose you constructed XCD where XCD is similar to XBA. Using
AB
.
the equal angles given by the angle-bisector, find CD
18 (Angle-bisector theorem). Find
BX
XC
in terms of triangle lengths.
What is another point on BC with an auspicious ratio of distances?
The midpoint U on BC forms the ratio BU
= 1. The lines connecting A,
UC
B, C to the midpoints and the lines connecting A, B, C to the feet of the
angle-bisectors seems to have something in common. What is it?
|
→. Let X, Y , Z be on BC, CA, AB. Suppose we know
do AX, BY , CZ concur?
19. Let BY , CZ meet at P . How can we turn
CY AZ
,
Y A ZB
20. Show that if AX meets BY , CZ at P as well, then
21. If
BX
XC
·
CY
YA
·
AZ
ZB
BX CY AZ
,
,
.
XC Y A ZB
When
in terms of P ?
BX
XC
·
CY
YA
·
AZ
ZB
= 1.
= 1, where must X be?
22 (Ceva’s theorem). Given X, Y , Z on segments BC, CA, AB, show that
BX CY
·
· AZ = 1 when AX, BY , CZ concur.
XC Y A ZB
|
→. What happens if BX
is the same, but X only has to be on line BC, and
XC
not on segment BC (i.e., not between B, C)?
23 (Menelaus’s theorem). Given X, Y , Z on lines BC, CA, AB, show that
BX CY
·
· AZ = 1 when X, Y , Z are collinear (lie on a line).
XC Y A ZB
These conditions on Ceva and Menelaus seem indistinguishable. In fact,
Menelaus’s theorem is slightly imprecise here. If only we can distinguish
them by assigning a sign, like a negative sign, on one of them. There is
actually a tool called directed lengths that can solve this issue. But for now,
look at the tool we used to prove these theorems.
April 8, 2016
Pearland MC
4
|
→. Find the area of a triangle given its sidelengths.
|
→. Split up the triangle into triangles that have the same height.
24. When a line is tangent to a circle, how must the line relate to the center
and point of tangency?
|
→. [P ] will denote the area of P , whatever P may be.
25. Show that [ABC] = rs, where r is the inradius (radius of the incircle)
(a = BC, b = CA, c = AB).
and s is the semi-perimeter a+b+c
2
|
→. How can we take advantage of the angle-bisector property and change it
into lengths? What does the incircle tell us about lengths?
26. Construct a special point so that there will be triangles similar to ARI,
BP I.
r
r
27. Find s−a
and s−b
in terms of a length involving the A-excenter of ABC
(doesn’t make sense unless you try to construct it the point in the previous
problem).
28 (Heron’s formula). Show that [ABC] =
q
s(s − a)(s − b)(s − c).
Exercises
29 (Isotomic conjugates). Let line AP meet BC at X. The isotomic conjugate of AP is the line AX 0 where BX = X 0 C. Show that for any point P
inside ABC, the isotomic conjugates of AP , BP , CP concur.
30 (Brian Liu, appeared on AoPS WOOT 2014–15 Practice Olympiad 2).
Cyclic hexagon RAN DOM is given such that line DR is perpendicular to
AM , RA = AN , and N D = DO. It is known that DR, M N , and AO
intersect at E. If a, b, c denote the areas of ARM ED, AN D, DOM E, respectively, then ab − b2 = 2b + c = 2015b. The result when the area of ARE
is subtracted from area of RM E is n. Find the remainder when n is divided
by 1000.
April 8, 2016
3
Pearland MC
5
Circles determined by triangles
Recall that the importance of perpendicular bisectors gave rise to our study
of perpendiculars (altitudes) and bisectors of segments (midpoints). We will
see that they are more closely related than that.
31 (Nine-point circle). For ABC, let its altitudes be AP , BQ, CR, intersecting at the orthocenter H. Let the midpoints of HA, HB, HC be T , U ,
V respectively. Let the midpoints of BC, CA, AB be X, Y , Z respectively.
How many points can you find lying on one circle (prove it in two ways)?
Are some points even diametrically opposite (form a diameter of the circle)?
|
→. One of the proofs relies on not explicitly constructing it, and not proving
immediately that 9 points lie on a circle. First, consider when 3 + 1 points
lie on a circle.
32. Let A, B, C lie on a circle centered at O. What does it mean, in terms
of angles, for A and C to lie on a circle centered at O?
33. Find 6 AOB in terms of 6 ACB.
34. Start with A, B. Given some x > 0, find all C where 6 ACB = x.
35. Find conditions for A, B, C, D to lie on a circle.
|
→. Recall that we used similar triangles when finding ratios. Can you clearly
define similarity?
|
→. Prove the existence of the nine-point circle using the circumcircle of
ABC. Where is the center?
Now, we’re going to prove it again, but with a perhaps more familiar tool.
36 (Thales of Miletus, 6th century BCE). Show that AB is the diameter of
a circle containing C when 6 ACB = 90◦ .
|
→. Which points lie on the circle with diameter T X?
37. Show that Y T , Y X are perpendicular. Show that U T , U X are perpendicular.
|
→. Which points lie on the circle with diameter U Y ?
38. Show that T , U , V , X, Y , Z lie on a circle.
April 8, 2016
Pearland MC
6
39. Show that P , Q, R lie on this circle as well.
|
→. Let the A-excenter, B-excenter, C-excenter of ABC be IA , IB , IC respectively. How does A relate to the triangle IA IB IC ?
40. Find 6 IA AIB .
41. Find the orthocenter of IA IB IC .
42. Describe the circumcircle of BIC.
43. Let R be the circumradius of IA IB IC and r be the radius of the nine-point
circle of ABC. Find Rr .
Exercises
44 (“Simson” line). For ABC, let its circumcircle contain P . Let the perpendiculars from P to BC, CA, AB have feet X, Y , Z. Show that X, Y , Z
lie on a line.
45 (Conway’s circle). For ABC, define A1 on AB opposite of B such that
AA1 = BC. In the same way, define A2 on AC. Analogously define B1 , B2 ,
C1 , C2 . Show that A1 , A2 , B1 , B2 , C1 , C2 lie on a circle. What is the radius?