## [幾何]幾何競賽題

### [幾何]幾何競賽題

1.Let P be a point inside the triangle ABC such thatAB=BC, ∠ABC=80° ,∠PAC=40° and ∠ACP=30°. Find ∠BPC.

2.D and E are points on sides BC and AC of a triangle ABC such that AD and BE are angle bisectors of the triangle ABC. If DE bisects ∠ADC, find ∠A.
追求神乎其技,至高無上的數學境界!~

### Re: [幾何]幾何競賽題

1.Let P be a point inside the triangle ABC such thatAB=BC, ∠ABC=80° ,∠PAC=40° and ∠ACP=30°. Find ∠BPC.

1.

BAC=BCA=50oCAP=40o，故AP之伸延線⊥BC；

☆子 是也

G@ry

3. In a trapezoid ABCD the sides BC and AD are parallel, AC=BC+AD, and the angle between the diagonals is equal to 60 degrees. Prove that AB=CD.

(剛剛出完才發現這題好像是秒殺題= =)
追求神乎其技,至高無上的數學境界!~

We draw a parallelogram BCKD using BD as a side
=>Let angle ACK=angle AKC= x
2x+180-x+120=360  =>x=60°
Hence we can find that both BOC and DOA are equilateral triangle
As a result , triangle BOA is congruent to COD
we can have a result which is BA=CD

Q.E.D.

I thought the problem you post were hard, so I took it to school to solve it.
Never did I know it was that easy! I thought there was a trick!
Atra esternī ono thelduin
Mor'ranr līfa unin hjarta onr
Un du evarīnya ono varda.

May good fortune rule over you
And the stars watch over you.

gkw0824usa

4.A triangle ABC has base AB = 1 and the altitude from C length h. What is the maximum possible product of the three altitudes? For which triangles is it achieved?

(如果有人看不懂,要中譯的話跟我說....)
追求神乎其技,至高無上的數學境界!~

2.D and E are points on sides BC and AC of a triangle ABC such that AD and BE are angle bisectors of the triangle ABC. If DE bisects ∠ADC, find ∠A.

F是三角形ABC的內心
G是三角形ACD的內心

∠BFC=90度+1/2*∠A=180度-1/4*∠A
3/4*∠A=90度
∠A=120度

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5.ABC is a triangle with incenter I. Show that the circumcenters of IAB, IBC, ICA lie on a circle whose center is the circumcenter of ABC.
追求神乎其技,至高無上的數學境界!~

5.ABC is a triangle with incenter I. Show that the circumcenters of IAB, IBC, ICA lie on a circle whose center is the circumcenter of ABC.

∠BIC=90度+∠A/2
∠BQC=360-2∠BIC=360-180-∠A=180-∠A
∠BOC=2∠A

BO=CO
∠BOC=2∠BQC

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4.A triangle ABC has base AB = 1 and the altitude from C length h. What is the maximum possible product of the three altitudes? For which triangles is it achieved?

h為定值→ab*sinC為定值
sinC值越大ab越小

h*h/(h^2+0.5^2)^1/2*h/(h^2+0.5^2)^1/2
=h^3/(h^2+0.5^2)

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6.The sides AB, BC, CD and DA of the quadrilateral ABCD are respectively  equal to the sides A'B', B'C', C'D' andD'A' of the quadrilateral A'B'C'D' and it is known that AB∥CD and B'C'∥D'A'.
Prove that both quadrilaterals are parallelograms.
追求神乎其技,至高無上的數學境界!~

Prove that both quadrilaterals are parallelograms.

☆子 是也

G@ry

TO：G@ry

http://blog.pixnet.net/ej0cl6
↑這是最近成立的數學BLOG
裡面有些幾何的東西
大家可以參觀看看呢

☆ ~ 幻 星 ~ ☆

☆ ~ 幻 星 ~ ☆ 寫到:TO：G@ry

☆子 是也

G@ry

追求神乎其技,至高無上的數學境界!~

7.The square PQRS is placed inside the square ABCD in such a way that the segments AP, BQ, CR and DS intersect neither each other nor the square PQRS. Prove that the sum of areas of quadrilaterals ABQP and CDSR is equal to the sum of the areas of quadrilaterals BCRQ and DAPS.
追求神乎其技,至高無上的數學境界!~