2. Let ABC be such that the sides AB and AC are not equal. Let 11 be the bisecto

Question

2. Let ABC be such that the sides AB and AC are not equal. Let 11 be the bisector of the angle ZBAC and let l2 be the perpendicular bisector of the side BC. Then l and l2 cannot be parallel and they cannot be the same line (why?). Now let D be the point wherel1 and /2 intersect. Show that A, B, C, and D all lie on the same circle. Hint First, consider the circle M that passes through A, B, and C. BC is a chord of M, so M intersects /2 in two points. Of these, let G be the one that lies on the side of BC opposite A. Now show that ZGAB-LGAC. To do this, use the fact that A, B, C, and G lie on a circle and that angles inscribed in a circle subtended by the same chord must be equal. Finally, conclude that G D. Next, let E be the foot of the perpendicular from D to AB and let F be the foot of the perpendicular from D to AC. Show that one of these two points lies on the triangle itself, but not both. Hint An easy way is to argue that if both lay on the triangle, or both lay outside the triangle, you could use the argument that we just did in class to show that AB- AC, which contradicts the assumption!

Explanation / Answer

Solution

perpendicular bisector CD of a line segment AB is a line segment perpendicular to AB and passing through the midpoint M of AB. The perpendicular bisector of a line segment can be constructed using a compass by drawing circles centered at A and B with radius AB and connecting their two intersections. This line segment crosses AB at the midpoint M of AB (middle figure). If the midpoint M is known, then the perpendicular bisector can be constructed by drawing a small auxiliary circle around M, then drawing an arc from each endpoint that crosses the line AB at the farthest intersection of the circle with the line (i.e., arcs with radii AA' and BB' respectively). Connecting the intersections of the arcs then gives the perpendicular bisector CD. Note that if the classical construction requirement that compasses be collapsible is dropped, then the auxiliary circle can be omitted and the rigid compass can be used to immediately draw the two arcs using any radius larger that half the length of AB. The perpendicular bisectors of a triangle A_1 A_2 A_3 are lines passing through the midpoint M_i of each side which are perpendicular to the given side. A triangle's three perpendicular bisectors meet at a point O known as the circumcenter, which is also the center of the triangle's circumcircle.

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