Winter 2009 MATH 330B/539 Geometry Homework 7 Hints
Assigned Homework Set #7
- Due Friday, March 6, 2009 at the start of class.
- 7.1 # 3, (5 omitted), 6, 8
Hints for Assigned Problems:
- Hints for #3: (Refer to Figure 7.1 on page 318.)
- Let O be the orthocenter of triangle(ABC). Then O is the intersection of the three perpendicular bisectors of the sides of triangle(ABC). These three bisectors go through midpoints M1, M2, and M3.
- The three medians of triangle(ABC) are concurrent at a point called the Center of the triangle. It is labeled C.
- One of these medians is line AM1.
- Consider triangle(ACH) and triangle(M1CO). We know that the three points A, C, and M1 are collinear, because they both lie on the median AM1. But we do not know whether or not the three points H, C, and O are collinear. We need to prove that they are collinear and that HC = 2CO.
- What do you know about the way that points N2 and H partition the altitude AF1?
- Note that the line M1O is an altitude of the medial triangle(M1M2M3). What do you know about the way that the point O partitions this altitude?
- Based on the two previous items, what can you say about the ratio AH/M1O?
- What do you know about the way location of point C on median AM1?
- Based on the two previous items, what can you say about the ratio AC/CM1?
- What can you say about angle(HAC) and angle(M1OC)?
- Based on the previous steps, what can you say about triangle(HAC) and triangle(M1OC)?
- Based on the previous step, what can you say about angle(ACH) and angle(M1CO)?
- Based on previous step, what can you say about the collinearity of points H, C, and O?
- What can you say about the spacing of points H, C, and O? Explain.
- Hint for #6: In Figure 2, extend the lines that trisect the sides until they meet at a point outside the triangle.
- Instructions for #8: Don't try to solve this problem. But do go online and find out information about it. Write a short (two or three paragraph) synopsis of the method of proof.
Suggested problems:
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Last updated March 8, 2009.
