An interesting question about triangle

 

Known: $ \ triangle {ABC} $ is a positive triangle, $ ad = be = CF $.

Proof: $ \ triangle {def} $ is a positive triangle.

 

Analysis:

Make the outer circle $ \ triangle {ABC} $ of $ \ odot {o} $, and extend $ ad, Be, CF $ and the outer side and the outer circle to two points respectively.

If the conclusion is true, it is equivalent to $ \ triangle {HCF}, \ triangle {Jad}, \ triangle {LBE}, \ triangle {BHD}, \ triangle {CJE }, \ triangle {alf} $ is a positive triangle.

Proof:

For quadrilateral $ abhc $,PtolemyTheorem knows,

$ Ah \ cdot BC = AB \ cdot CH + bH \ cdot AC \ rightarrow Ah = BH + Ch $.

Similarly, $ bj = AJ + cj, \ Cl = Al + BL $.

In $ \ triangle {HCF} $, $ \ angle {FHC} =\ angle {ABC} = 60 ^ {\ CIRC} $. the relationship between $ Ch $ and $ CF $ is discussed below.

If $ ch <CF \ rightarrow ch <Ad \ rightarrow BH> DH $.

On the other hand, by $ \ angle {CFH} <\ angle {CHF} = 60 ^ {\ CIRC} <\ angle {HCF} \ rightarrow ch <cf <HF $.

Available: $ \ angle {LFA} <\ angle {alf} <\ angle {FAL} \ rightarrow Al <AF <LF \ rightarrow BL> CF = be \ rightarrow $

$ CJ> Ce> je \ rightarrow AJ <be = Ad \ rightarrow BH <BD <DH $.

Conflict with $ BH> DH $!

Similarly, $ ch> CF $ is not valid.

Therefore, $ \ triangle {HCF} $ is a positive triangle.

Similarly, $ \ triangle {Jad}, \ triangle {LBE} $ are all positive triangles. Therefore, $ \ triangle {def} $ is a positive triangle.

Q $ \ cdot $ e $ \ cdot $ d

An interesting question about triangle