![]() Prove the hint is valid before they can use it in their solution. A right triangle has one right angle, and an obtuse triangle has one obtuse angle. My geometry course and my number theory course, that if a student usesĪ hint given by the author of the text, then the student must first In an acute triangle all angles are acute. One minor point: nowhere in the book (that I could find) was thisįact, or material relating to it, presented. ![]() This certainly gave a solution, except for Hint was for the student to use the fact that the altitudeīisected the angle. Problem to see why he required the cevian lines to be altitudes. One based on the properties ofĬeva's and Menelaus' theorems and the other using the properties of aĪfter I completed these solutions, I looked at Eves' hint for the Property 4: The circumcenter and the orthocenter of an obtuse-angled triangle lie outside the triangle. Subtracting the above two, we have, 2 + 3 < 90. Solution was valid for any concurrent cevian linesįact I found two nice solutions. We know that by angle sum property, the sum of the angles of a triangle is 180. The only property of altitudes that I used was theįact that the altitudes of a triangle are concurrent. This sounded like a nice problem, but when I worked it in preparationįor class, I discovered that I did not use the fact that The dotted red lines in the figures above. The term altitude is often used interchangeably with 'height.' Below are a few examples depicting the altitudes of some geometric figures. If are the feet of the altitudes on the sides In geometry, the altitude of a geometric figure generally refers to a perpendicular distance measured from the base of the figure to its opposite side. In Howard Eves' book, College Geometry, I found the following problem: ![]() Some students in my geometry class were strugglinig with the concepts of cross ratio and harmonic division, so I went looking for some problems to work for them as examples of using these concepts. It appears that the three altitudes of the triangle once again intersect in a single point, however this time they intersect.
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