Math 5335: hints and supplementary exercises

Math 5-335 Fall 2002
Hints and supplementary exercises

Hint for §2.5; problem 30 In the first part of the problem, we're supposed to plot the line consisting of points whose points have barycentric coordinates (r,s,t) with r = -1. We can draw such a line by finding two points on it. The two easiest such points to find are the ones where the line r = -1 intersects the lines and . By Theorem 16 in Chapter 2, those two lines are characterized by the equations t = 0 and s = 0 respectively. The points therefore are (-1,2,0) = -A +2B = B - (A - B) and (-1,0,2) = -A +2C = C - (A - C).

Supplementary exercise #1 Consider the same triangle ABC as in problem 30 of §2.5.

Sketch the line joining A to the point (0,¹/₃,²/₃)

(which lies on the line

Show that for every point (r,s,t)

of this line, the equality of ratios ^s/_t = ¹/₂ holds.

Show that, more generally, if P = (0,b,c)

is a point of the line

, then for every point (r,s,t)

of the line joining A and P, the equality of ratios ^s/_t = ^b/_c holds.

Hint for §3.12; problem 59 Here's a diagram of the setup:

If you follow the plan presented in the text (and also in the lecture on Thursday, October 9), then we're given that D and E are on the line through C parallel to

and further that A and E are on opposite sides of

while B and D are on opposite sides of

. Under this setup, we need to prove that

and

are opposite rays, so that we can apply Proposition 34 of Chapter 1 [about complementary angles] and also Proposition 33 of Chapter 1 [about addition of angular measure]. Anyway, this is what's the main unresolved issue, since things were set up to make it fairly easy to verify the hypotheses of the proposition about "alternate interior angles" [Proposition 36 of Chapter 1]. Well that's the question is, and it's "sort of clear" what kind of tools are available. But it seems hard to come up with clear suggestions about how to do it gracefully.

Here's an alternate version of the setup which is quite acceptable, and actually quite compatible with the general approach of our text. [I looked this up somewhere; I'll divulge the identity of my source at some later date . . . ] The idea is to define E by saying that its coordinate vector is given as E =C + (B - A) and similarly to define D by saying that its coordinate vector is given as D =C + (A - B). This guarantees immediately that we have opposite rays [do you see the reason for this/], and it's also not hard to see that the hypotheses of the proposition about alternate interior angles are verified. (This is a small additional price that we have to pay for using this alternate approach, but it seems much more manageable than the proof of oppositeness of the rays in the other approach.) Oh yes, there's one other thing that musn't be neglected, either under this approach or under the other approach. Namely, we have to show [for instance] that the ray is in the interior of the angle CBD, in order to apply the result about angle addition. Under our approach it isn't too difficult. It may require some thinking, but if you go back to the definition of interior of an angle (and maybe another related definition), it is possible to do something that's actually pretty convincing.

Hint for §3.12; problem 68 Let (r,s,t) be the orthocenter. The question is to find explicit formulas for r, s and t.

Theorem 6 of Chapter 3 gives explicit formulas for the barycentric coordinates of the feet of the 3 altitudes. So, we can apply the result of Supplementary exercise #1 to find formulas for the ratios ^r/_s , ^s/_t , and ^r/_t . This most likely will produce quotients of quadratic polynomials in a, b and c.

One slightly displeasing aspect of our formulas for these ratios is that different denominators appear in the formulas for ^s/_t and ^r/_t . But it may be possible to do something to get two fractions with the same denominator -- possibly some polynomial that is of degree 4 in a, b and c. In a similar way, you may be able to add some appropriate "padding" to your formula for ^r/_s .

These "augmented" formulas might sugggest some coherent guesses about possible polynomial values for r, s and t. One small difficulty is that we may not have r + s + t = 1. But if not, then by what should we divide everything in order to get a triple in which the sum of the coordinates is = 1?? Do you recognize the sum of your three 4^th degree polynomials as something that may have occurred in one of our proofs??

Supplementary exercise #2 Given triangle ABC, let P be the midpoint of , let Q be the midpoint of , and let R be the midpoint of .

Show that is parallel to .
Suggestion: Using appropriate coordinate vectors for P and Q, find a direction indicator for and compare it with one of the usual direction indicators for .
Similarly, show that is parallel to and that is parallel to .
Using the vector formulas developed on parts a and b -- or by any other valid method -- verify the following equalities of lengths of segments:
- || = ¹/₂ ||
- || = ¹/₂ ||
- || = ¹/₂ ||
Show that the circumcenter of triangle ABC is equal to the orthocenter of triangle PQR.

Supplementary exercise #3 Given triangle ABC and triangle PQR. Assume that there exists a nonzero real number v such that:
|| = v || || = v || and || = v ||.

Show that if the orthocenter of triangle ABC is (r,s,t)^ABC, then the orthocenter of triangle PQR is (r,s,t)^PQR. ( We use the notations ABC and PQR as part of the superscript to indicate the triangle relative to which we are taking the barycentric coordinates . . . )

Suggestion: Refer to the formula from Theorem 7 of Chapter 3 of the text. What power of v can be factored from the numerators and from the denominators?

Supplementary exercise #4 Let triangle ABC and triangle PQR be as in Supplementary exercise #2, and let (r,s,t) be the orthocenter of triangle ABC.

Use the results of supplementary exercises 2 and 3 to find a formula for the barycentric coordinates of the circumcenter of triangle ABC (in terms of r, s, and t).
Suggestion: Consider the given information as being a formula for the coordinate vector of the orthocenter, namely rA + sB + tC. Working with the other results we have obtained, try to develop a formula for the coordinate vector of the circumcenter.

Use the result of part a to show that the orthocenter, the circumcenter, and the centroid of triangle ABC are collinear.

Comments and questions top; roberts@math.umn.edu

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