It is known that in △ABC, O is the external heart, I is the internal heart, AB+AC = 2bc. Verification: OI⊥AI (figure).

Inspection center: the judgment and nature of similar triangles; Theorem of circle angle; The inscribed circle and heart of a triangle.

Special topic: proving the problem.

Because I is the heart, AC/CE = AB/BE = AI/IE (AC+AB)/BE = AB/BE. And because AC+BC=2BC, AB=2BE .. From △ABE∽△ADC, we know that AD=2DC. , and DC=DI (intrinsic essence).

∴，AC/ce = ab/be = ai/ie(AC+ab)/be = ab/be。

∫AC+AB = 2BC，

∴ AB = 2be。 From △ Abbe ·∴△ADC, we know that AD=2DC. ..

∵DC=DI (intrinsic nature),

∴AD=2DI.