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BC . sin(~ABC), PABC = 2AB· BC . cos(~ABC). 85 (The Herron-Qin Formula9 ) In triangle ABC. 2 16SABC ~ = 4AB ==2 . CB - 2 P ABC ' Proof. 84, sin(~ABC) = ~~~Zg, cos(~ABC) sinUABC)2 + cos(~ABC)2 = 1, we have Thus 2 16SABC -2-2 = 4AB . ) is an ancient Chinese mathematician who discovered the area fonnula for triangles in exact fonn of this proposition [169). 86 The oriented angle between two directed lines PQ and AB. denoted by 4(PQ, AB). is defined as follows. Take points O , X , and Y such that OYQP and OX BA are parallelograms.

3. 6. Show that 1~ + ~~ + g; = 2. 4. Let ABC D be a quadrilateral and 0 a point. Let E. F, G, and H be the intersections of lines AO, BO, CO, and DO with the corresponding diagonals BD, AC. BD, and AC al Show that HcFAwGi'i AHCFBE7JG_1 of the quad n'l ater. - . 5. 17, let L 1 , L 2 , L3 be the intersections of lines OP and CCl> lines OQ and BBl> and lines OS and AA 1 • Show that LI, L 2 , L3 are collinear. 22 Let AB and CD be two non-degenerate lines. If AB and CD do not have any common point, we say that AB is parallel to CD.

Show that PC = 2P B. Proof. Note that LP AC = LAM C and LP AB LAC M . By the co-side and co-angle theorems, PC PB = = \lPAC \lPAC \lMAC \lPAB = \lMAC \lPAB PA·AC AC · MC MA · MC PA · AB AC . AC = AB· AB = AB MA · AB MA·AB MA = 2. 38 AM is the median of triangle ABC. D, E are points on AB, AC such that AD = AE. DE and AM meet in N. Show that ~~ = 1~ · Chapter 1. 39 Four rays passing through a point 0 meet two lines sequentially in A, B, C, D and P, Q, R , S _Show that ~~:~g = ~~:~_ Proof By the co-side and co-angle theorem = = AB-CD·PS-QR AD·BC PQ-RS AB CD PS QR - -- -- - AD BC PQ RS \lOAB _\lOCD _\lOPS _\lOQS \lOAD \lOBC \lOPQ \lORS \lOAB \lOCD \lOPS \lOQR \lOPQ - \lORS - \lOAD - \lOBC OA·OB-OC -OD-OP-OS-OQ-OR OP · OQ -OR -OS · OA -OD·OB-OC = 1.

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