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Geometry Problem. Post your solution in the comment box below.Level: Mathematics Education, High School, Honors Geometry, College.Details: Click on the figure below.
Problem 1312The extension of OH intersects AB and BC in K,L respectively .According to the problem 761 the triangle BKL is equilateral .Is <AOC=2<ABC=120, <AHC=90+<ABC/2=120.So AOHC is cyclic.Draw CPM//KL (M in AB, P in AH).Is <AMC=120=<AOC or AMOHC is , so AH-HC=AH-HP=AP=12.Draw MN perpendicular AP. Is OD=MN=x, MA=2x andtriangle MAN from pythagorean theorem 4x^2=x^2+36 or x=2√(3 ).APOSTOLIS MANOLOUDIS KORYDALLOS PIRAEUS GREECE
Extend AH to meet BC at EThe perpendicular from O to BC cuts BC in half and say meet at P. So OPED is a rectangle with PE = OD = x -------(1)let AB = c and consider the 30-60-90 triangle ABEwe have BE = c/2 and AE = Sqrt(3)c/2 -------(2)From (1) and (2) EC = PC-PE = BP-PE = BE-PE-PE = c-4x/2 --------(3)Now consider the 30-60-90 triangle ECHwe have HE = (c-4x)/2Sqrt(3)CH = (c-4x)/Sqrt(3) --------(4)Given AH-CH = 12=> (AE-HE)-CH = 12=> Sqrt(3)c/2-(c-4x)/2Sqrt(3)- (c-4x)/Sqrt(3) = 12 (from (2) and (4))=> 3c-c+8x-2c+4x = 12*2*Sqrt(3) => x = 2Sqrt(3)
Let R be the circumradius.Since < AOC = < AHC = 120, AOHC is concyclic. So < OHD = 30 and OH = 2xApplying Ptolemy,2x.b + R. CH = R. AHBut R = b/sqrt3 considering Tr. AOC. So x = (AH - CH).R/b = (12 / 2)/sqrt3x = 2.sqrt3Sumith PeirisMoratuwaSri Lanka
Extend AH to P (P on circle) => ΔHPC equilateralExtend CH to M (M on circle) => ΔABC ≡ ΔAMC so if we draw perpendicular to CM and AB we get x inradius of AHG (G on AB)AD = 1/2 AP => AD = DH+HP = DH+HC (1) But AH-HC = 12 (2) From (1) and (2) DH = 6 From ΔODH and pythagore theorem x = 2√3
https://goo.gl/photos/Mo9Wudh9iSaMpX4S8Let c= AH, a= HCLet AH cut circle ABC at ENote that BC is the perpen. Bisector of HE => HCE is equilateralSo HE=HC=aD is the midpoint of AE so AD= ½(a+c)AOK is 30-60-90 triangle so AO= AC/sqrt(3)In triangle AHC we have angle AHC= 120 and AC^2= a^2+c^2+a.cIn triangle AOD => x^2= AO^2-AD^2X^2= 1/3(a^2+c^2+a.c)-1/4(a+c)^2After simplification we get 12.x^2= (a-c)^2= 144So x=2.sqrt(3)
Sumith Peiris, Moratuwa, Sri Lanka- This solution is EXCELLENT!