[數學]IMO98 Problem 1

[數學]IMO98 Problem 1

--- 於 星期一 五月 12, 2003 4:38 pm


IMO98 Problem 1

A convex quadrilateral ABCD has perpendicular diagonals.
(凸四邊形ABCD對角線互相垂直)

The perpendicular bisectors of AB and CD meet at a unique point P inside ABCD.
(AB & CD 之2中垂線交於ABCD內一點P)

Prove that ABCD is cyclic if and only if triangles ABP and CDP have equal areas.
(請證明:ABCD共園,若且唯若三角形ABP,CDP面積相等)

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Raceleader 於 星期一 五月 12, 2003 4:42 pm


Let AC and BD meet at X. Let H, K be the feet of the perpendiculars from P to AC, BD respectively. We wish to express the areas of ABP and CDP in terms of more tractable triangles. There are essentially two different configurations possible. In the first, we have area PAB = area ABX + area PAX + area PBX, and area PCD = area CDX - area PCX - area PDX. So if the areas being equal is equivalent to: area ABX - area CDX + area PAX + area PCX + area PBX + area PDX. ABX and CDX are right-angled, so we may write their areas as AX.BX/2 and CX.DX/2. We may also put AX = AM - MX = AM - PN, BX = BN - PM, CX = CM + PN, DX = DN + PM. The other triangles combine in pairs to give area ACP + area BDP = AC.PM + BD.PN. This leads, after some cancellation to AM.BN = CM.DN. There is a similar configuration with the roles of AB and CD reversed.

The second configuration is area PAB = area ABX + area PAX - PBX, area PCD = area CDX + area PDX - area PCX. In this case AX = AM + PN, BX = BN - PM, CX = CM - PN, DX = DN + PM. But we end up with the same result: AM.BN = CM.DN.

Now if ABCD is cyclic, then it follows immediately that P is the center of the circumcircle and AM = CM, BN = DN. Hence the areas of PAB and PCD are equal.

Conversely, suppose the areas are equal. If PA > PC, then AM > CM. But since PA = PB and PC = PD (by construction), PB > PD, so BN > DN. So AM.BN > CM.DN. Contradiction. So PA is not greater than PC. Similarly it cannot be less. Hence PA = PC. But that implies PA = PB = PC = PD, so ABCD is cyclic.

http://www.kalva.demon.co.uk/imo/isoln/isoln981.html

Raceleader
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--- 於 星期一 五月 12, 2003 8:20 pm


徵求解析解法, meowth給答者$2000. ^_^

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