PROPOSITION XXXVIII. PROBLEM. 238. To describe a circumference through three points not in the same straight line. . В Draw A B and BC. Bisect A B and BC. At the points of bisection, E and F, erect is intersecting at 0. From ( as a centre, with a radius equal to 0 A, describe a circle. O ABC is the O required. For, the point o, being in the I EO erected at the middle of the line A B, is at equal distances from A and B; and also, being in the I FO erected at the middle of the line C B, is at equal distances from B and C, § 58 (every point in the I erected at the middle of a straight line is at equal distances from the extremities of that line). and a O described from ( as a centre, with a radius equal to 0 A, will pass through the points A, B, and C. Q. E. F. · 239. SCHOLIUM. The same construction serves to describe a circumference which shall pass through the three vertices of a triangle, that is, to circumscribe a circle about a given triangle. PROPOSITION XXXIX. PROBLEM. 240. Through a given point to draw a tangent to a given circle. с м Fig. 2. Α Η CASE 1. — When the given point is on the circumference. point on the circumference. From the centre 0, draw the radius OC. Then C M is the tangent required, $ 186 (a straight line I to a radius at its extremity is tangent to the O). CASE 2. — When the given point is without the circumference. Let A B C (Fig. 2) be the given circle, 0 its centre, E the given point without the circumference. It is required to draw a tangent to the circle A B C from the point E. Join () E. On 0 E as a diameter, describe a circumference intersecting the given circumference at the points M and H. . Draw 0 M and 0 H, E M and E H. § 204 § 186 (a straight line I to a radius at its extremity is tangent to the O). In like manner we may prove H E tangent to the given O. Q. E. F. 241. COROLLARY. Two tangents drawn from the same point to a circle are equal. PROPOSITION XL. PROBLEM. Let ABC be the given triangle. Draw the line A E, bisecting 2 A, Draw E HI to the line A C. The O KH M is the required. and EM I to BC. Iden. Cons. ..A A KE= A AHE, $ 110 (Two rt. Á are equal if the hypotenuse and an acute Zof the one be equal respectively to the hypotenuse and an acute 2 of the other). .. EK= EH, (being homologous sides of equal A). .. EK, E H, and E M are all equal. .. a O described from E as a centre, with a radius equal to E H, will touch the sides of the A at points H, K, and M, and be inscribed in the A. § 174 Q. E. F. PROPOSITION XLI. PROBLEM. 243. Upon a given straight line, to describe a segment which shall contain a given angle. E Let A B be the given line, and M the given angle. It is required to describe a segment upon the line A B, which shall contain < M. At the point B construct Z A B E equal to Z M. Bisect the line A B at F, and erect the I FH. From the point B, draw BO I to E B. From 0, the point of intersection of FH and B O, as a centre, with a radius equal to 0 B, describe a circumference. Now the point 0, being in a I erected at the middle of A B, is at equal distances from A and B, § 58 (every point in a I erected at the middle of a straight line is at equal dis tances from the extremities of that line) ; .. the circumference will pass through A. Cons. ..BE is tangent to the O, § 186 (a straight line I to a radius at its extremity is tangent to the O). .. L A B E is measured by 1 arc A B, $ 209 (being an Z formed by a tangent and a chord). Also any Z inscribed in the segment A H B, as for instance ZA K B, is measured by ļ arc A B, (being an inscribed Z). § 203 .:. LAKB= L A B E, ..LAKB= 2 M. Q. E. F. PROPOSITION XLII. PROBLEM. 244. To find the ratio of two commensurable straight lines. E H Let A B and C D be two straight lines. It is required to find the greatest common measure of A B and C D, so as to express their ratio in numbers. Apply C D to A B as many times as possible. Suppose twice with a remainder E B. Suppose three times with a remainder F D. Suppose once with a remainder H B. Suppose once with a remainder K D. Suppose K D is contained just twice in H B.. The measure of each line, referred to K D as a unit, will then be as follows: — HB = 2 KD; "CD – 18 K D' .. the ratio of A B _ 41 •• the ratio of CD18 Q. E. F. |