(b) Show that a quadrilateral ABCD is a parallelogram if and only if AB DC and AB = DC.

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1 Homework 1 Geometry 461 ue: Monday Sep 8, In the figure, is isosceles, with =, and angle bisectors and have been drawn. rove that =. 2. In the figure, and are isosceles triangles that share base. Show that the line joining the vertices of these two triangles is perpendicular to the base. 3. Recall that by definition, a quadrilateral is a parallelogram if and. In other words, a quadrilateral is a parallelogram if its opposite sides are parallel. (a) Show that a quadrilateral is a parallelogram if and only if = and =. (b) Show that a quadrilateral is a parallelogram if and only if and =. 4. In the diagram of roblem 1, do not assume the given information of that problem, but assume instead that and are altitudes of. (Recall that this means that is perpendicular to and is perpendicular to.) Show that = if and only if =. 5. Recall that by definition, a rectangle is a parallelogram all of whose angles are 90 and note that if one of the angles of a parallelogram is a right angle, then the other three angles must also be right angles. Show that a parallelogram is a rectangle if and only if its diagonals are equal.

2 Homework 2 Geometry 461 ue: Monday Sep 15, Medians and are drawn in. (a) Show that. (b) Show that = if and only if =. HINT: For (a), use areas to show that the perpendicular distances from and to line are equal. The if direction of (b) is fairly easy, but the only if direction seems difficult. Here too you can use the fact that the perpendicular distances from and to are equal. 2. In the figure, sides and of quadrilateral are parallel. (Recall that a quadrilateral with one pair of opposite sides parallel is called a trapezoid.) Show that = if and only if =. HINT: raw a line through parallel to. 3. In the figure, we started with, and we found a point O, equidistant from the three vertices of the triangle. (s we shall see, there always is such a point.) Next, we reflected point O in the three sides of O the triangle to obtain points, and. (To reflect O in side, for example, we dropped a perpendicular from O to, and then we took point to be the point on this perpendicular line on the other side of and at the same distance from as the original point O.) Finally, we drew. rove that = and prove that the corresponding sides of these two triangles are parallel. HINT: Show that quadrilateral O is a rhombus. 4. If we draw two medians of a triangle, we see that the interior of the triangle is divided into four pieces: three triangles and a quadrilateral. rove that two of these small triangles have equal areas, and show that the other small triangle has the same area as the quadrilateral.

3 Homework 3 Geometry 461 ue: Monday Sep 22, ircles centered at and intersect at points and as shown, and diameters and are drawn. rove that points, and are collinear. (Show, in other words, that they lie on a common line.) 2. oints,, and are chosen arbitrarily on a circle, as shown, and points W,, and are the midpoints of {, {, {{{{ and {, { respectively. Show that W and are perpendicular. W 3. Two tangents to a circle centered at O are drawn from a point outside of the circle. If the points of tangency are denoted S and T, show that O bisects S T and that S = T. 4. In the figure, is a common chord of two circles, and also is the bisector of RS, where R is a point on one circle and S is a point on the other. oint U is where chord R of the left circle meets the right circle, and similarly, V is where chord S of the right circle meets the left circle. rove that RU = SV. R U S V

4 Homework 4 Geometry 461 ue: Monday Sep 29, point is chosen on the base of isosceles, and then perpendiculars and are dropped to sides and, respectively. (Note that it may be necessary to extend side or to do this.) In the figure, we show two possible positions for, where the second one is labelled.) Show that as moves along line segment, the sum + remains constant. HINT: Use areas. 2. Let be any quadrilateral, and let W,, and be the midpoints of sides,, and, respectively. Show that W is a parallelogram. ' 3. Two points and are chosen on a circle, and then two more circles are drawn: the circle through centered at and the circle through centered at. The two new circles intersect at a point inside the original circle, as shown. lso, the circle centered at meets the original circle at a point (and also at, of course). Line is then drawn, meeting the original circle at point. rove that { { = 60. NOTE: The result of this problem implies that distance is the radius of the original circle. Explain why. 4. The diagonals of quadrilateral are drawn as shown in the figure, and we find that =. rove that =. HINT: Two proofs are possible; one shows that is a cyclic quadrilateral and the other uses similar triangles.

5 Homework 5 Geometry 461 ue: Monday Oct 6, oints and are selected on two sides of, as shown, and segments and are drawn. Then and are drawn parallel to and, respectively. Show that. 2. Let be a point outside of a circle centered at point O, and let be the midpoint of O. Show that the locus of all midpoints of segments, where is on the given circle, is a circle centered at. NOTE: It is not actually necessary to assume that lies outside of the given circle, but we make that assumption for convenience. 3. uadrilateral is inscribed in a circle, and its opposite sides are extended to meet at points and, as shown. rove that the bisectors of and are perpendicular. HINT: Let be the point where the two bisectors meet and let be a point on the bisector of, to the left of the circle. Show that = be expressing each of these angles in terms of arcs. 4. In the figure, point is tangent to the circle at and the midpoint M of secant lies on the circle. If length M = 1, compute length and prove that your answer is correct. M

6 Homework 6 Geometry 461 ue: Monday Oct 13, points per problem 1. Suppose that M is a median of. Show that M is an altitude of the triangle if and only if it bisects. 2. In the figure, the upper circle is centered at point, which lies on the lower circle, and the two circles intersect at points and. hord of the upper circle is drawn, and it meets the lower circle at point. rove that =. HINT: How do, and compare? 3. In the figure, vertices and of are joined to points and on the opposite sides, and lines and meet at point. Suppose that = (2/3) and = (2/3). rove that and are medians of. 4. In the figure, point H is the orthocenter of. This point is reflected in the three sides of the triangle to obtain points, and. Show that all three angle bisectors of go through the point H. H HINT: Recall that we know that points, and all lie on the circumcircle of.

7 Homework 7 Geometry 461 ue: Monday Oct 20, Given, the sides are extended and three circles are drawn, as shown, each tangent to one side of the triangle and to two extensions of sides. (These are called the exscribed circles of the triangle.) oints, and are the centers of the exscribed circles. Show that is the pedal triangle of. HINT: Use roblem 3 of Homework Show that there is no point inside such that every line through cuts the triangle into two pieces of equal area. HINT: Show that if there were such a point, it would have to lie on each median of the triangle. 3. Suppose that the Euler line of is perpendicular to side. rove that =. 4. The purpose of this problem is to give another proof of the fact that the altitudes of a triangle are always concurrent, and so you should not use any facts that you already know about altitudes or the orthocenter. Let be any triangle and draw a line through each vertex parallel to the opposite side, thus forming, as shown. Show that the altitudes of are the perpendicular bisectors of the sides of the new. NOTE: Since we know that the perpendicular bisectors of the sides of any

8 Homework 8 Geometry 461 ue: Monday Oct 27, In the figure, V W = V U and W V = W U. rove that line U bisects W UV. HINT: Reflect point U in sides and. 2. In the situation of roblem 1, assume also that UW = UV. Show that UV W is the pedal triangle of. W U V 3. Given with orthocenter H and circumcenter O, let be the Euler point on line H and let M be the midpoint of side. ssume that, H, M and O are four different points that form a genuine quadrilateral. (There are some degenerate cases where this does not happen.) (a) Show that quadrilateral HM O is a parallelogram. (b) Show that (H) 2 = 4R 2 a 2, where as usual, a =. HINT: For (a), show that the diagonals of HMO bisect each other. NOTE: If = 90, then H and are the same point and if =, then all four of, H, M and O lie on the altitude from. In all other cases, HMO is a genuine quadrilateral. The formula of part (b) remains true even in the degenerate cases. 4. Show that for every triangle, the sum of the squares of the lengths of the medians is three quarters of the sum of the squares of the lengths of the sides. 5. quadrilateral with side lengths 2, 3, 2 and 4 is inscribed in a circle, as shown. Find the radius of the circle and justify your answer

9 Homework 9 Geometry 461 ue: Monday Nov 3, In the figure, the side of is trisected by points R and S. Similarly, T and U trisect side and V and W trisect side. Each vertex of is joined to the two trisection points on the opposite side, and the intersections of these trisecting lines determine, as shown. rove that the sides of are parallel to the sides of. HINT: rove that = (3/5)R. V W R U S T 2. Let,, and be any four points on a circle. Let W be the orthocenter of, let be the orthocenter of, let be the orthocenter of and let be the orthocenter of. rove that the four line segments W,, and all have the same midpoint. W HINT: It is enough to show that W and have the same midpoint, and for this, it suffices to show that W is a parallelogram. Use HW7 roblem 3(b) for this. NOTE: It is true, in fact, that the point that is the midpoint of all four line segments also lies on the nine point circles of all four triangles. 3. Let I be the incenter of and let, and, respectively, be the points (other than, and ) on the circumcircle of where lines I, I and I meet the circle. Show that I is the orthocenter of 4. Suppose is a right triangle. Show that s = r + 2R, where r, s and R have their usual meanings.

10 Homework 10 Geometry 461 ue: Monday Nov 10, Let I be the incenter of. If I is also the incenter of the medial triangle of, show that is equilateral. 2. Let be an interior point of equilateral. If perpendiculars, and are dropped to the sides of the triangle, as shown, show that the quantity + + is a constant, independent of the choice of point. 3. Suppose that = 90 in and let be the altitude of this right triangle. (a) Express the lengths, and in terms of the lengths a, b and c of the sides of. (b) Let r be the inradius of and write r 1 and r 2 to denote the inradii of and, respectively. rove that r 1 + r 2 + r =. HINT: Similar triangles can be used for (a) and roblem 4 of HW 8 is relevant for (b). 4. oints W,, and are the midpoints of the sides of quadrilateral as shown, and is the intersection of W with. Two of the four small quadrilaterals are shaded. Show that is the midpoint of both W and and that the shaded area is exactly half of the area of quadrilateral. W HINT: For the second part, decompose the whole area into four triangles

11 Homework 11 Geometry 461 ue: Monday Nov 17, Let F be the Fermat point of, where each of the angles of the triangle is less than 120. (a) Show that F = F = F = 120. (b) If F is the incenter of, prove that the triangle is equilateral. (c) If F is the orthocenter of, show that the triangle is equilateral. 2. In the figure, line segments, and EF are common tangents to two given circles and the six endpoints of these segments lie on the circles, as shown. rove that the midpoints, and of the three segments are collinear. E F 3. If we join the three vertices of to the three points where the incircle is tangent to the sides, we know that the three lines thus obtained are concurrent. The point of concurrence is called the Gergonne point of the triangle. Show that if the Gergonne point of is the incenter of the triangle, then the triangle is equilateral. 4. oints and are chosen on two sides of, as shown, and lines and meet at. Show that lies on the median from vertex if and only if.

12 Homework 12 Geometry 461 ue: Monday Nov 24, Suppose, and R are concurrent evians in. onsider the three lines obtained by joining the midpoint of each evian to the midpoint of the corresponding side. (For example, the midpoint of is joined to the midpoint of side.) Show that these lines are concurrent. HINT: Show that the midpoints of the evians lie on the sides of the medial triangle. 2. Given, there is a way to get new evian lines from old ones: reflect in the angle bisector. For example, in the figure, is a evian from vertex and is the bisector of. The new evian was drawn so that is the bisector of, or equivalently, so that =. ' Now suppose, and R are concurrent evians in. rove that the three new evians obtained by reflecting the three given ones in the three angle bisectors are also concurrent. NOTE: The reflections of the medians of a triangle are called the symmedians and their point of concurrence is the Lemoine point of the triangle. 3. Given an acute angled triangle, show that the point of concurrence of the reflections of the three altitudes is the circumcenter of the triangle. (The word reflection here refers to the process of the previous problem.) 4. Given, let be the point 1/3 of the way from to, as shown. Similarly, is the point 1/3 of the way from to and lies 1/3 of the way from to. In this way, we have constructed a new triangle, starting with an arbitrary triangle. Now apply the same procedure to, thereby creating. Show that the sides of are parallel to the (appropriate) sides of. What fraction of the area of is the area of? ' " ' " " '

13 Homework 13 Geometry 461 ue: Monday ec 1, points per problem 1. Equilateral is inscribed in a circle, and point is chosen on the circle, as shown. rove that = +. HINT: Extend line segment past to a point chosen so that =. 2. In the figure, is the common chord of two intersecting circles and points and are chosen, one on each circle. Line also meets the circles at points and, as shown. ssuming that the midpoint M of lies on, show that M is also the midpoint of. M 3. The accompanying diagram was drawn in the following way. Starting with, points and were chosen to be the trisection points of side, and point on side was constructed one fifth of the way from to. Then lines and were drawn, meeting at point, and line was drawn, meeting at W. Finally, W was drawn. rove that W is parallel to. W 4. In the figure, and EF G are squares sharing vertex. oint M is the midpoint of G and line segment M was extended to meet E at. rove that E is perpendicular to M and that E = 2M. M G E F