PROBLEMS AND APPLICATIONS. 1. Given two roads of different width at right angles to each other, to connect them by a road whose sides are arcs of circles tangent to the sides of the roads. (a) Make the construction shown in the figure and prove that it has the required properties. Ca C1 (b) Is this construction possible when the given roads are not at right angles to each other? Illustrate. (c) Can the curve be made long or short at will? (d) Make the construction if the given roads have the same width. 2. Two circles C and C' are tangent at the point D. AB is a segment through D terminating in the circles. Prove that the radii CA and C'B are parallel. 3. Through a point on the bisector of an angle to construct a circle tangent to both sides of the angle. = EP and CONSTRUCTION. Through the given point P draw EP 1 to AP. Lay off ED: at D construct DC LAD meeting the line AP in C. Then C is the center of the required circle and CD is its radius. PROOF: Draw PD and prove that A DEP is isosceles and hence also A PDC. B B Is it possible to construct another circle having the properties required? If so, construct it. frequent occurrence in decorative work such as the steel ceiling panel given here. 4. In an isosceles triangle construct three circles as shown in the figure. SUGGESTION. First construct the inscribed circle with center O. Let the bisector of A meet this circle in a point P. Then use Ex. 3. B 5. The angles formed by a chord and a tangent are equal respectively to the angles inscribed in the arcs into which the end-points of the chord divide the circle. 6. If a triangle whose angles are 48°, 56°, and 76° is circumscribed about a circle, find the number of degrees in the arcs into which the points of tangency divide the circle. 7. Divide each side of an equilateral triangle into three equal parts (Ex. 5, § 159) and connect points as shown in the figure. Prove that DEFGHK is a regular hexagon. 8. If a circle is inscribed in the triangle of Ex. 7, prove that all sides of the hexagon are tangent to the circle. SUGGESTION. Show that the perpendicular bisectors of the segments HK, KD, DE meet in a point A equidistant from these segments. 9. Within a given square construct' four equal circles so that each circle is tangent to one side of the square and to two of the circles. SUGGESTION. First construct the diagonals of the square. 10. In the figure, ABCD is a rectangle D D B H G K D E with AD = AB. E (a) Prove that these quadrant arcs are tangent to each other in pairs and also to the semicircles. (b) Lines are drawn tangent to the arcs at the points where these are met by the diagonals of the squares AEFD and BCFE. Prove that these lines form squares KLMN and XYZW. (c) Construct the small circles within each of these squares. The above design occurs in fan-vaulted ceilings. The gothic or pointed arch plays a conspicuous part in modern architecture, and examples of it may be found in almost any city. Its most common use is in church windows. The figure represents a so-called equilateral gothic arch. The arcs AC and BC are drawn from B and A as centers respectively, and with AB as a radius. The segment AB is called the span of the arch, and the point C its apex. A struction and prove that the figure has the required properties. (b) Prove that DE and DF are tangent to each other. Also BF and BC, and AE and AC. (c) What axis of symmetry has this figure? 12. A triangle ABC whose angles are 45°, 80°, and 55° is inscribed in a circle. Find the angles of the triangle formed by the tangents at A, B, and C. 13. Inscribe a circle in an equilateral gothic arch ABC. SUGGESTIONS. Construct CDL to AB and extend it to P, making DP = AB. From P construct a tangent to AC at L. (a) Prove that ABDPA BLP and hence PLBD. (b) A OLPA BDO and hence OD = OL. Then O OD is the required circle. See § 209. L spect to the line PD, and hence if the circle is proved tangent to AC, we know at once that it is tangent to BC. 14. In the figure ABC is an equilateral gothic arch with a circle inscribed, as in Ex. 13. (a) Construct the two equilateral arches GHE and HKF, as shown in the figure. CONSTRUCTION. Draw BK and AGLAB. With a radius equal to OD + DB, and with 04 as center draw arcs meeting BK and AG in K and G respectively. Draw GK, construct the G arches and show that each is tangent to the circle. (b) Do the points E and F lie on the circle? SUGGESTION. Suppose KF to be drawn, and compare HKF with ZHKO by comparing the sides HK and KF and also GK and KO. 15. Construct an arc passing through a given point B, and tangent to a given line AD at a given point D. 16. In the figure ABC is an equilateral arch. BK is of BD. 5 KBF and AHE are equal equilateral arches. Arcs KQ and HQ are tangent to arcs KF and HE respectively. C M HD K B D H A B From Lincoln Cathedral, England. B K (a) Find by construction the center O of the circle tangent to AC, BC, KF, and HE, and give proof. (b) Find by construction the centers of the arcs KQ and HQ. How is this problem related to Ex. 15? 17. Two circles are tangent to each other internally. Find the locus of the centers of all circles tangent to both externally. 18. Two circles are tangent to each other externally. Find the locus of the centers of all circles tangent to both, but external to one and internal to the other. 19. Two equal circles are tangent to each other externally. Find the locus of the centers of all circles tangent to both. 20. AD'B is an angle whose vertex is outside the circle and whose sides meet the circle in the points A and B, while ADB is an inscribed angle intercepting the arc AB. Prove that ZADB><AD'B, provided each of the segments D'A and D'B cuts the circle at a second point. 21. Through two given points A and B construct a circle tangent to a given line which is perpendicular to the line AB. Is this construction possible if the given line passes through either of the points A or B? If it meets AB between these points? 22. In kicking a goal after a touchdown in the game of football, the ball is brought back into the field at right angles to the line marking the end of the field. The distance between the goal posts being given, and also the point at which the touchdown is made, find by a geometrical construction how far back into the field the ball must be brought in order that the goal posts may subtend the greatest possible angle. 808 D' EA Touch down Kick D D Goal posts B B |