Geometry Problem. Post your solution in the comments box below.

Level: Mathematics Education, High School, Honors Geometry, College.

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## Wednesday, November 26, 2014

### Geometry Problem 1063: Triangle, Orthocenter, Altitudes, Equal Product of the Lengths of Segments

Labels:
altitude,
orthocenter,
product,
triangle

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Since ABA₁B₁ is concyclic, with AB as diameter,

ReplyDeleteAA₁ and BB₁ are two chords intersecting at H,

thus AH×HA₁ = BH×HB₁.

Similarly consider cyclic quadrilateral BCB₁C₁,

we have BH×HB₁ = CH×HC₁.

Hence, AH×HA₁ = BH×HB₁ = CH×HC₁.

triunghiul ANB1este asemenea cu triunghiul BHA1,BHC1~CHB1 (UU)=>AH/BH=HB1/HA1;HB/HC=HC1/HB1 => AH×HA₁ = BH×HB₁ = CH×HC₁.

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