Find Area of Rhombhus In Triangle ABC

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In triangle ABC, the task is to find points X, Y, and Z on sides AB, BC, and CA, respectively, to form a rhombus AXYZ and demonstrate that its area is less than or equal to half the area of triangle ABC. Participants emphasize the importance of showing prior attempts to solve the problem to receive effective guidance. One user explored using trapeziums but struggled to progress, later recognizing that AY represents angle bisectors and XZ serves as the perpendicular bisector. They noted the need to establish relationships between the segments, specifically focusing on ratios involving AB, AC, BY, and CY. The discussion highlights the necessity of understanding geometric properties to advance in solving the problem.
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In triangle ABC, find points X, Y and Z on AB, BC and CA such that AXYZ is a rhombhus. Show that area of the rhombhus AXYZ<= 1/2A(ABC)
 
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You have posted a number of questions without any indication what, if anything, you have done on it yourself. We will not do your homework for you! We will try to give hints but we need to know what you have tried so we can gauge your understanding and see where you are going wrong.
 
i tried with the trapezium BXZY and CYXZ. but couldn't reah nowhere. after that a close look made me understand thatAY is the angle bisectors and XZ is its pependicular bisector. but i don't know how to use this property to move on. but i wrote all ratios i willget like- AB/AC = BY/CY. but the need is AX + AZ + 1/2(AB + AC)
 

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