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**Extra resources for Challenging Problems in Geometry (Dover Books on Mathematics)**

Use Theorem #55c. Ptolemy’s Theorem isn't really utilized in this technique. (Note: There are circumstances to be thought of. ) 8-1 strategy I: Draw a line via C, parallel to AB, assembly PQR at D. turn out that ΔDCR ~ ΔQBR, and ΔPDC ~ ΔPQA. process II: Draw , the place M, N, and L are on . turn out that ΔBMQ ~ ΔANQ, ΔLCP ~ ΔNAP, and ΔMRB ~ ΔLRC. 8-2 process I: examine the components of some of the triangles shaped, which proportion a similar altitude. (Note: There are circumstances to be thought of. ) approach II: Draw a line via A, parallel to BC, assembly CP at S, and BP at R. turn out that ΔAMR ~ ΔCMB, ΔBNC ~ ΔANS, ΔCLP ~ ΔSAP, and ΔBLP ~ ΔRAP. (Note: There are instances to be thought of. ) process III: Draw a line via A and a line via C parallel to BP, assembly CP and AP at S and R, respectively. turn out that ΔASN ~ ΔBPN, and ΔBPL ~ ΔCRL; additionally use Theorem #49. (Note: There are situations to be thought of. ) procedure IV: reflect on BPM a transversal of ΔACL and CPN a transversal of ΔALB. Then practice Menelaus’ Theorem. 8-3 practice Ceva’s Theorem. 8-4 Use similarity, then Ceva’s Theorem. 8-5 Use Theorem #47; then use Ceva’s Theorem. 8-6 Use Theorem #47; then use Menelaus’ Theorem. 8-7 Use Theorem #47; then use Menelaus’ Theorem. 8-8 First use Ceva’s Theorem to discover BS; then use Menelaus’ Theorem to discover TB. 8-9 Use Menelaus’ Theorem; then use Theorem #54. 8-10 Use either Ceva’s and Menelaus’ Theorems. 8-11 think about NGP a transversal of ΔAKC, and GMP a transversal of ΔAKB. Then use Menelaus’ Theorem. 8-12 Draw , and , the place D and E lie on BC. For either elements (a) and (b), neither Ceva’s Theorem nor Menelaus’ Theorem is used. organize proportions concerning line segments and parts of triangles. 8-13 expand FE to fulfill at P. contemplate AM as a transversal of ΔPFC and ΔPEB; then use Menelaus’ Theorem. 8-14 Use one of many secondary effects proven within the answer of challenge 8-2, strategy I. (See III, IV, and V. ) Neither Ceva’s Theorem nor Menelaus’ Theorem is used. 8-15 Use Menelaus’ Theorem and similarity. 8-16 Use Menelaus’ Theorem, taking KLP and MNP as transversals of ΔABC and ΔADC, respectively the place P is the intersection of AC and LN. 8-17 Use Theorems #36, #38, #48, and #53, by means of Menelaus’ Theorem. 8-18 Taking RSP and R′S′P′ as transversals of ΔABC, use Menelaus’ Theorem. additionally use Theorems #52 and #53. 8-19 think about RNH, PLJ, and MQI transversals of ΔABC; use Menelaus’ Theorem. Then use Ceva’s Theorem. 8-20 Use Ceva’s Theorem and Theorem #54. 8-21 Draw traces of facilities and radii. Use Theorem #49 and Menelaus’ Theorem. 8-22 Use Theorems #48, #46, and Menelaus’ Theorem. 8-23 Use Menelaus’ Theorem solely. 8-24 (a) Use Menelaus’ Theorem and Theorem #34. (b) Use Menelaus’ Theorem, or use Desargues’ Theorem (Problem 8-23). 8-25 expand DR and DQ via R and Q to fulfill a line via C parallel to AB, at issues G and H, respectively. Use Theorem #48, Ceva’s Theorem and Theorem #10. additionally turn out . 8-26 technique I: Use the results of challenge 8-25, Theorem #47, and Menelaus’ Theorem. procedure II: Use Desargues’ Theorem (Problem 8-23).