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i solved the question no. 2.. please see the solution and tell me whether it is correct or not :
in triangle ABC , FM = 2 and MC = 4
thus FM/MC = 1/2
also BE which is the median passes through M.
Thus we conclude that M is the centroid of triangle ABC.
thus as AD is perpendicular to BC and CF is angle bisector of triangle ABC, therefore triangle ABC is equilateral.
thus in triangle ACF ,by Pythagoras theorem we have
AB^2 = 36 + AB^2/4
or AB = 12/3^1/2
THUS PERIMETER OF TRIANGLE ABC = 12 TIMES ROOT OF 3
Well, problem 2 has many solutions i think as i got 2 solutions. The first one requires to use the formula of the length of the angle bisector repeatedly. This will finally fetch you that △ABC is equilateral. That is the aim of the problem actually. Prove that △ABC is equilateral. If u cannot do just post a comment, Il type out the detailed solution for you.
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This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
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Top NewestHint for (1): Draw the line ℓ parallel to BC through A. Suposse DE and DF meet ℓ at P and Q.
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thanks
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Solution to #1 We drop FX⊥BC and EY⊥BC. We need to show that
∠FDA=∠ADE
⟺ ∠FDX=∠EDY
⟺ tan∠FDX=tan∠EDY
⟺ DXFX=DYEY ... (i) . Thus if we can prove that (i) is true, we can claim that AD bisects ∠FDE
Now, Ceva's theorem in △ABC we have
DCBD×EACE×FBAF=1.... (ii)
Now, △FBX∼△ABD and △ECY∼△ACD and so we have after some computation, FX=BDBX×AD and EY=CDCY×AD.
Therefore, EYFX=DB×CYBX×CD... (iii) Coming back to ii) and putting EACE=YDCY and FBAF=BXDX we have
DCBD×EACE×FBAF=1
⟹ DCBD×YDCY×XBDX=1
⟹ DYDX=CY×BDBX×DC ....(iv). Equating (iii) and (iv), we have
DYDX=EYFX and thus we have DXFX=DYEY. Thus we proved (i) and hence we are done! :)
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i solved the question no. 2.. please see the solution and tell me whether it is correct or not :
in triangle ABC , FM = 2 and MC = 4 thus FM/MC = 1/2 also BE which is the median passes through M. Thus we conclude that M is the centroid of triangle ABC. thus as AD is perpendicular to BC and CF is angle bisector of triangle ABC, therefore triangle ABC is equilateral. thus in triangle ACF ,by Pythagoras theorem we have AB^2 = 36 + AB^2/4 or AB = 12/3^1/2 THUS PERIMETER OF TRIANGLE ABC = 12 TIMES ROOT OF 3
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correct!
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very clever. Better than mine solution.
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One more hint prove triangle QAP AND PAD similar By using cevians theorem and parallel lines
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Well, problem 2 has many solutions i think as i got 2 solutions. The first one requires to use the formula of the length of the angle bisector repeatedly. This will finally fetch you that △ABC is equilateral. That is the aim of the problem actually. Prove that △ABC is equilateral. If u cannot do just post a comment, Il type out the detailed solution for you.
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My new problem I created due to my misread of Q2
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by the way megha did you qualified for INMO?
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yes and not megha, "Megh"
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Problem 1 is really simple if u know the concept of harmonic bundles. The official solution is completely unmotivated
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