Construction of Triangles, etc.

127. Construct an isosceles triangle which shall have the vertical angle equal to four times an angle at the base.

128. ABCD is a parallelogram and E is a point in AB; construct on AE as base, and with EAD as one of its angles, a parallelogram equal to ABCD.

129. Construct an isosceles triangle which shall have one-third of each of the angles at the base equal to half the vertical angle.

130. Construct a right-angled triangle, having given its perimeter and one of the acute angles.

131. Construct a ▲ ABC, having given

(1) AB, AC and the altitude from A;
(2) B, C and the altitude from A;

(3) 4B, BC and the median from A;

(4) BC, CA and the angle between BC and the median from A.

132. Construct a triangle, having given the perimeter, the altitude from the vertex, and one angle at the base.

133. Construct a ▲ ABC, having given BC+CA-AB, ▲ B, ≤ C. (See Miscellaneous Deduction 27, page 186.)

134. Construct a ▲ ABC, in which AN is the altitude from A, and AK is the bisector of the

A, having given

(1) BN, BK, ▲ B;

(2) BK, CK, 4 B;

(3) CK, 4 B, 4 C;

(4) BK, CK, ≤ B ~ ≤ C;

(5) AK, BK, ▲ B ~ ≤ C;

(6) BN, BK, } < A+ ≤ B ;

(7) AB, AK, }< A+ < C.

135. Construct a rectangle ABCD, having given

(1) AB and AC;

(2) AB+BC and AB - BC;

(3) AC and AB+BC;

(4) AC and AB ~BC;

(5) AB and AC+BC;

(6) AB and AC-BC.

136. Construct a ABCD, whose diagonals intersect in O, having given

(1) AB, AC+BC, ▲ A;

(2) AB, AC~BC, ▲ &;

(8) AC, AB+BC, ▲ A;

(4) AC, AB-BC, ▲ A;
(5) AB, AC+BD, ‹ AOB;
(6) AB, AC~BD, ▲ AOB.

137. Construct a rhombus ABCD, having given

(1) AB and AC;

(2) AB and AC+BD;

(3) AB and AC-BD;
(4) AC+BD and ▲ A;
(5) AB+ AC and A.

138. Construct a parallelogram, such that two of the sides shall lie on two given parallel straight lines, the other two sides shall pass through two given points, and the area shall equal a given

area.

139. If A be a given point, BC and DE two given parallel straight lines, construct a square APQR, so that

(1) P and Q may lie on BC and DE;

(2) P and R may lie on BC and DE.

140. Construct a trapezium, having given one vertex, the midpoints of the diagonals, and the mid-point of one of the parallel sides.

141. Construct a quadrilateral, having given two opposite sides, the diagonals, and the angle contained by the diagonals.

142. Inscribe a square in the space enclosed by two equal intersecting circles.

143. In a given square inscribe a square whose area shall be three-fourths of the area of the given square.

144. Describe a square whose four sides shall pass respectively through four given points.

Show that, in general, six squares may be so described if the four points may be taken in any order.

Consider separately the cases in which the four points form a parallelogram, a rectangle, a rhombus, and a square.

SS 31-41. EUCLID II.

Theorems.

145. ABC is a triangle, right-angled at A, and AC or CA is produced to E, so that AC AE=AB2; show that AC'CE=BC2.

146. If P be any point within a ▲ ABC, and PL, PM, PN be drawn BC, CA, AB; show that

BC2+CA2+AB2=2(BC. BL+CA·CM+AB.AN).

147. A, B, C, D, E are five points in order on the same straight line, and AB=CD; show that AE DE+BD.CD=BE.CE.

148. If a straight line be divided into any number of segments, the square on the whole line is equal to the squares on the segments, together with twice the rectangles contained by each pair of segments.

149. K is any point on the diagonal AC of a rectangle ABCD; EKF parallel to BC meets AB, CD in E, F, and GKH parallel to AB meets AD, BC in G, H; show that

150. A, B, C, D are four points in order in the same straight line; show that AD2+BC2=AC2+2AB CD+BD2.

151. A, B, C, D, E are five points in order in the same straight line, and AB=BC, CD=DE; show that AD2+3DE2=BE2+3AB2.

152. ABC is a triangle, right-angled at A, and AN1BC; show that
(BC+AN)2=(AB+AC)2+AN2.

153. ABC is a triangle, right-angled at A; show that
(BC+CA+AB)2=2 (BC+CA) (AB+BC).

154. ABCD is a square; P and Q are two points in BD; show that
AP2~AQ2=PQ (PB~QD).

155. Of all parallelograms with equal perimeters, the square has the greatest area.

156. ABCD is a square; O is the mid-point of BD and P is any other point in BD; OM, PN are perpendicular to AB; show that AM2 > AN PN and that ▲ AOM >^ APN.

157. A, B, C, D are four points in order in the same straight linə, and AB=BC; show that AD2-BD2=3AB2+2AB CD.

158. ABCD is a quadrilateral, right-angled at B and D. If AB+AD> BC+CD, show that AB~AD<BC~CD.

159. In the ▲ ABC, AN is the altitude from A, and P is any point in AN; show that AB~AC<PB~PC.

160. In the ▲ ABC, AK is the bisector of the A, and P is any point in AK; show that

AB-AC> PB~PC.

161. ABCD is a crossed quadrilateral in which ABCD and AD1BC; E, F, G, H are the mid-points of AB, CD, AD, BC; show that AB2 CD2=4EF.GH.

162. AB is bisected in O and divided in P; show that

AP2+PB2=2 AP· PB+4 OP2.

163. In the figure of § 36, Ex. 1, show that

(1) AH> BH, H'A> AB;

(2) AB2=H'A AH=H'B· HB;

(3) AB2=2HB2+3AH·HB=2H'B2-3H'A⚫H'B;

(4) (AB+BH)2=5AH2; (AB+H'B)2=5H'A2.

164. AB is divided in H, so that AB HB=AH2, and produced to K, so that BK AH; show that AK BK=AB2.

=

165. ABC is a triangle, right-angled at A, and AN is perpendicular to BC. If AC=BN, prove that BC is divided in medial section.

166. If, in the figure of § 36, Ex. 1, an equilateral triangle PAH be described, show that PB=AG.

167. In any right-angled triangle, twice the sum of the squares on the medians is equal to three times the squale on the hypotenuse.

168. The base BC of a ▲ ABC is trisected in D and E; show that

AB2+AC2=AD2+AE2+4DE2.

169. ABC is a triangle, right-angled at A, and BC is trisected in D and E; show that

3(DE2+EA2+AD2)=2BC2.

170. In ▲ ABC, AB > AC, BC is trisected in D and E; show that AB2-AC2-3 (AD2-AE2).

171. ABCD is a quadrilateral; F, H are the mid-points of BC, DA, and I, J are the mid-points of AC, BD; show that

AB2+CD2=2 FH2+2 IJ2.

172. A is a given point and BX a given straight line; P and Q are two points in BX such that AP2+BP2=AQ2+BQ2; if C be the mid-point of AB, show that CP=CQ.

173. Show that the triangle is acute-angled whose sides measure 50, 41, and 39 inches respectively, and find the length of the altitude drawn to the shortest side.

કો 174. Show that the triangle is obtuse-angled, whose sides measure 52, 33, and 25 inches respectively, and find the length of the altitude drawn to the intermediate side.

175. A, B, C, D are four points in order on the same straight line, E is the mid-point of AB, G of CD, and O of EG; if P be any other point in the line, show that

PA2+PB2+PC2+PD2=OA2+OB2+OC2+OD2+4PO2.

176. E, F, G, H are the mid-points of AB, BC, CD, DA, the sides of a quadrilateral ABCD; show that

AB2+CD2+2EG2=BC2+DA2+2FH2.

177. In the ▲ ABC, E is the mid-point of AC and AN1BC; show that BC BN=BE2~CE2.

178. G is the point of intersection of the medians of the ▲ ABC, and P is any other point, show that

3(PA2+PB2+PC2)=BC2+CA2+AB2+9PG2.

179. ABCD is a quadrilateral; E is the mid-point of AB, G of CD O of EG and P is any point; show that

PA2+PB2+PC2+PD2=OA2+OB2+OC2+OD2+4PO2.

180. ABC is a triangle right-angled at A, and BC=2AC; D is a point in AC produced, such that CD=BC; show that BD2=3BC2. 181. ABC is an equilateral triangle in which BC is produced to D, so that CD=2BC; show that AD2=7AB2.

$$ 39-41. EUCLID II.

Problems.

182. Construct a square equal to one-third of a given square. 183. Construct a rectangle equal to a given regular hexagon. 184. Construct a rectangle equal to a given regular octagon. 185. Construct two lines, having given their sum and the area of the rectangle they contain.