这是三角形的什么心？

hujunhua

问题：如图，若点O是其基于△ABC的Ceva三角形△DEF的外心，那么它是△ABC的什么心？
在三角形中心大全网站应该查得到，但需要计算出6.9.13三角形中该点的坐标，我一时计算不出来，查不了。
注：给定△ABC及平面上一点O，各顶点A, B, C通过O投影到对边所得的3点构成的三角形称为点O基于△ABC的Ceva三角形。

nyy

首先需要证明 点 O 能够成为Ceva三角形△DEF的外心，然后才能问这是什么心。

mathe

存在性很容易理解。如果我们只要求OE=OD, 那么，对于任意给定的E, 当D移动到B点时:OD=0<OE, 而当D移动到C点时，OD=EC>0=OE,
所以其中必然存在一个位置使得OE=OD. 唯一遗憾的是E可能无法唯一确定D,也就是对于给定的E可能可以有多个可选的D.

另外一方面，我们我们保持OE=OD, 将E移动到非常接近C时，OD总是很短，所以，OE也要很短，O点非常靠近C, 这时OF必然大于OE=OD.
但是另外一方面，当E移动到非常接近A,可以看出这时应该有BE~=AB, AD~=AB,AO+BO~=AB,所以要求O接近AB中点，OF~=0 <OE=OD.
所以移动E并且保持OE=OD的过程，必然存在某个E使得OF也等于它们。但是不知道能否保证唯一性

hujunhua

设`O`的面积坐标为`(x:y:z), x+y+z=1,O=xA+yB+zC`
那么$D={yB+zC}/{y+z},E={zC+xA}/{z+x},F={xA+yB}/{x+y}$
$OD={x}/{y+z}(yAB+zAC),\to |OD|^2={x^2}/{(y+z)^2}(c^2y^2+b^2z^2+(b^2+c^2-a^2)yz)$
轮换变量可得 $|OE|^2={y^2}/{(z+x)^2}(a^2z^2+c^2x^2+(c^2+a^2-b^2)zx)$
                    $|OF|^2={z^2}/{(x+y)^2}(b^2x^2+a^2y^2+(a^2+b^2-c^2)xy)$
列方程 $|OD|^2=|OE|^2=|OF|^2,x+y+z=1$没解出来。
不过得到`a=13,b=6,c=9`的数值解`x= 0.53611, y = 0.197125 , z = 0.266764`
不知道通过这个数值解在那个网站能不能查询到它的编号。

hujunhua

@wayne 为什么$D={yB+zC}/{y+z}$?

`O=xA+yB+zC` → `O-xA=yB+zC`→${O-xA}/{1-x}={yB+zC}/{y+z}$
${O-xA}/{1-x}$和${yB+zC}/{y+z}$的系数和都等于1，所以分别表示直线OA和直线BC上的点，两者相等，故为同一点，即两直线的交点`D`.

wayne

数值解不好对不上.

1. Block[{a=13,b=6,c=9},NSolve[{x+y+z==1,(x^2 (c^2 y^2+(-a^2+b^2+c^2) y z+b^2 z^2))/(y+z)^2==(y^2 (c^2 x^2+(a^2-b^2+c^2) x z+a^2 z^2))/(x+z)^2==((b^2 x^2+(a^2+b^2-c^2) x y+a^2 y^2) z^2)/(x+y)^2},{x,y,z,k}]]

复制代码

creasson

不是已知的心,  不妨设B=0, C=1, A=z, O=w, 并记$\bar{z}$为z的共轭， 则w是7次方程的解:
\[-z^8-4 w z^7+4 \bar{z} z^7+4 z^7+34 w^2 z^6-6 \bar{z}^2 z^6-28 w z^6+16 w \bar{z} z^6-16 \bar{z} z^6-6 z^6-20 w^3 z^5+4 \bar{z}^3 z^5-52 w^2 z^5-64 w^3 \bar{z}^2 z^5+96 w^2 \bar{z}^2 z^5-56 w \bar{z}^2 z^5+24 \bar{z}^2 z^5+68 w z^5+64 w^3 \bar{z} z^5-136 w^2 \bar{z} z^5+48 w \bar{z} z^5+24 \bar{z} z^5+4 z^5-233 w^4 z^4-\bar{z}^4 z^4+492 w^3 z^4+128 w^3 \bar{z}^3 z^4-192 w^2 \bar{z}^3 z^4+80 w \bar{z}^3 z^4-16 \bar{z}^3 z^4-222 w^2 z^4-320 w^4 \bar{z}^2 z^4+608 w^3 \bar{z}^2 z^4-308 w^2 \bar{z}^2 z^4+56 w \bar{z}^2 z^4-36 \bar{z}^2 z^4-36 w z^4+320 w^4 \bar{z} z^4-688 w^3 \bar{z} z^4+528 w^2 \bar{z} z^4-144 w \bar{z} z^4-16 \bar{z} z^4-z^4+416 w^5 z^3-736 w^4 z^3-64 w^3 \bar{z}^4 z^3+96 w^2 \bar{z}^4 z^3-36 w \bar{z}^4 z^3+4 \bar{z}^4 z^3+224 w^3 z^3+128 w^4 \bar{z}^3 z^3-384 w^3 \bar{z}^3 z^3+376 w^2 \bar{z}^3 z^3-144 w \bar{z}^3 z^3+24 \bar{z}^3 z^3+96 w^2 z^3+1344 w^5 \bar{z}^2 z^3-2912 w^4 \bar{z}^2 z^3+1928 w^3 \bar{z}^2 z^3-440 w^2 \bar{z}^2 z^3+56 w \bar{z}^2 z^3+24 \bar{z}^2 z^3-1344 w^5 \bar{z} z^3+3108 w^4 \bar{z} z^3-2224 w^3 \bar{z} z^3+376 w^2 \bar{z} z^3+80 w \bar{z} z^3+4 \bar{z} z^3-192 w^6 z^2+320 w^5 z^2-64 w^4 z^2-64 w^4 \bar{z}^4 z^2+224 w^3 \bar{z}^4 z^2-222 w^2 \bar{z}^4 z^2+68 w \bar{z}^4 z^2-6 \bar{z}^4 z^2-64 w^3 z^2-1152 w^5 \bar{z}^3 z^2+2816 w^4 \bar{z}^3 z^2-2224 w^3 \bar{z}^3 z^2+528 w^2 \bar{z}^3 z^2+48 w \bar{z}^3 z^2-16 \bar{z}^3 z^2-1472 w^6 \bar{z}^2 z^2+4128 w^5 \bar{z}^2 z^2-4214 w^4 \bar{z}^2 z^2+1928 w^3 \bar{z}^2 z^2-308 w^2 \bar{z}^2 z^2-56 w \bar{z}^2 z^2-6 \bar{z}^2 z^2+1472 w^6 \bar{z} z^2-3712 w^5 \bar{z} z^2+2816 w^4 \bar{z} z^2-384 w^3 \bar{z} z^2-192 w^2 \bar{z} z^2+320 w^5 \bar{z}^4 z-736 w^4 \bar{z}^4 z+492 w^3 \bar{z}^4 z-52 w^2 \bar{z}^4 z-28 w \bar{z}^4 z+4 \bar{z}^4 z+1408 w^6 \bar{z}^3 z-3712 w^5 \bar{z}^3 z+3108 w^4 \bar{z}^3 z-688 w^3 \bar{z}^3 z-136 w^2 \bar{z}^3 z+16 w \bar{z}^3 z+4 \bar{z}^3 z+512 w^7 \bar{z}^2 z-2432 w^6 \bar{z}^2 z+4128 w^5 \bar{z}^2 z-2912 w^4 \bar{z}^2 z+608 w^3 \bar{z}^2 z+96 w^2 \bar{z}^2 z-512 w^7 \bar{z} z+1408 w^6 \bar{z} z-1152 w^5 \bar{z} z+128 w^4 \bar{z} z+128 w^3 \bar{z} z-192 w^6 \bar{z}^4+416 w^5 \bar{z}^4-233 w^4 \bar{z}^4-20 w^3 \bar{z}^4+34 w^2 \bar{z}^4-4 w \bar{z}^4-\bar{z}^4-512 w^7 \bar{z}^3+1472 w^6 \bar{z}^3-1344 w^5 \bar{z}^3+320 w^4 \bar{z}^3+64 w^3 \bar{z}^3+512 w^7 \bar{z}^2-1472 w^6 \bar{z}^2+1344 w^5 \bar{z}^2-320 w^4 \bar{z}^2-64 w^3 \bar{z}^2 = 0\]

对于所给边长为9,6,13的示例, 解为
Root[49831302877314416640000 + 197787546356707786752000 #1 +
  24085371910008912320563200 #1^2 - 58797906987239838687476480 #1^3 +
  63363649185560099859496169 #1^4 - 39543087908417049628502896 #1^5 +
  15793979873578280116526892 #1^6 - 4229345563892421513414640 #1^7 +
  779446413863812794409414 #1^8 - 100119874435084817968720 #1^9 +
  8952316765400348795468 #1^10 - 546772164890225790672 #1^11 +
  21770260705861194969 #1^12 - 509725700810202240 #1^13 +
  5328222552656640 #1^14, 8]

wayne

$R^2 =|OD|^2=|OE|^2=|OF|^2$是关于k的7次方程的根. 代码是

1. GroebnerBasis[{x+y+z==1,x^2 (c^2 y^2+(-a^2+b^2+c^2) y z+b^2 z^2)==k (y+z)^2,y^2 (c^2 x^2+(a^2-b^2+c^2) x z+a^2 z^2)==k (x+z)^2,(b^2 x^2+(a^2+b^2-c^2) x y+a^2 y^2) z^2==k (x+y)^2},{},{x,y,z}]

复制代码

1. a^24-12 a^22 b^2+66 a^20 b^4-220 a^18 b^6+495 a^16 b^8-792 a^14 b^10+924 a^12 b^12-792 a^10 b^14+495 a^8 b^16-220 a^6 b^18+66 a^4 b^20-12 a^2 b^22+b^24-12 a^22 c^2+108 a^20 b^2 c^2-420 a^18 b^4 c^2+900 a^16 b^6 c^2-1080 a^14 b^8 c^2+504 a^12 b^10 c^2+504 a^10 b^12 c^2-1080 a^8 b^14 c^2+900 a^6 b^16 c^2-420 a^4 b^18 c^2+108 a^2 b^20 c^2-12 b^22 c^2+66 a^20 c^4-420 a^18 b^2 c^4+1050 a^16 b^4 c^4-1200 a^14 b^6 c^4+420 a^12 b^8 c^4+168 a^10 b^10 c^4+420 a^8 b^12 c^4-1200 a^6 b^14 c^4+1050 a^4 b^16 c^4-420 a^2 b^18 c^4+66 b^20 c^4-220 a^18 c^6+900 a^16 b^2 c^6-1200 a^14 b^4 c^6+400 a^12 b^6 c^6+120 a^10 b^8 c^6+120 a^8 b^10 c^6+400 a^6 b^12 c^6-1200 a^4 b^14 c^6+900 a^2 b^16 c^6-220 b^18 c^6+495 a^16 c^8-1080 a^14 b^2 c^8+420 a^12 b^4 c^8+120 a^10 b^6 c^8+90 a^8 b^8 c^8+120 a^6 b^10 c^8+420 a^4 b^12 c^8-1080 a^2 b^14 c^8+495 b^16 c^8-792 a^14 c^10+504 a^12 b^2 c^10+168 a^10 b^4 c^10+120 a^8 b^6 c^10+120 a^6 b^8 c^10+168 a^4 b^10 c^10+504 a^2 b^12 c^10-792 b^14 c^10+924 a^12 c^12+504 a^10 b^2 c^12+420 a^8 b^4 c^12+400 a^6 b^6 c^12+420 a^4 b^8 c^12+504 a^2 b^10 c^12+924 b^12 c^12-792 a^10 c^14-1080 a^8 b^2 c^14-1200 a^6 b^4 c^14-1200 a^4 b^6 c^14-1080 a^2 b^8 c^14-792 b^10 c^14+495 a^8 c^16+900 a^6 b^2 c^16+1050 a^4 b^4 c^16+900 a^2 b^6 c^16+495 b^8 c^16-220 a^6 c^18-420 a^4 b^2 c^18-420 a^2 b^4 c^18-220 b^6 c^18+66 a^4 c^20+108 a^2 b^2 c^20+66 b^4 c^20-12 a^2 c^22-12 b^2 c^22+c^24-16 a^22 k+144 a^20 b^2 k-560 a^18 b^4 k+1200 a^16 b^6 k-1440 a^14 b^8 k+672 a^12 b^10 k+672 a^10 b^12 k-1440 a^8 b^14 k+1200 a^6 b^16 k-560 a^4 b^18 k+144 a^2 b^20 k-16 b^22 k+144 a^20 c^2 k-800 a^18 b^2 c^2 k+1360 a^16 b^4 c^2 k+640 a^14 b^6 c^2 k-5600 a^12 b^8 c^2 k+8512 a^10 b^10 c^2 k-5600 a^8 b^12 c^2 k+640 a^6 b^14 c^2 k+1360 a^4 b^16 c^2 k-800 a^2 b^18 c^2 k+144 b^20 c^2 k-560 a^18 c^4 k+1360 a^16 b^2 c^4 k+1600 a^14 b^4 c^4 k-7360 a^12 b^6 c^4 k+4960 a^10 b^8 c^4 k+4960 a^8 b^10 c^4 k-7360 a^6 b^12 c^4 k+1600 a^4 b^14 c^4 k+1360 a^2 b^16 c^4 k-560 b^18 c^4 k+1200 a^16 c^6 k+640 a^14 b^2 c^6 k-7360 a^12 b^4 c^6 k+4480 a^10 b^6 c^6 k+2080 a^8 b^8 c^6 k+4480 a^6 b^10 c^6 k-7360 a^4 b^12 c^6 k+640 a^2 b^14 c^6 k+1200 b^16 c^6 k-1440 a^14 c^8 k-5600 a^12 b^2 c^8 k+4960 a^10 b^4 c^8 k+2080 a^8 b^6 c^8 k+2080 a^6 b^8 c^8 k+4960 a^4 b^10 c^8 k-5600 a^2 b^12 c^8 k-1440 b^14 c^8 k+672 a^12 c^10 k+8512 a^10 b^2 c^10 k+4960 a^8 b^4 c^10 k+4480 a^6 b^6 c^10 k+4960 a^4 b^8 c^10 k+8512 a^2 b^10 c^10 k+672 b^12 c^10 k+672 a^10 c^12 k-5600 a^8 b^2 c^12 k-7360 a^6 b^4 c^12 k-7360 a^4 b^6 c^12 k-5600 a^2 b^8 c^12 k+672 b^10 c^12 k-1440 a^8 c^14 k+640 a^6 b^2 c^14 k+1600 a^4 b^4 c^14 k+640 a^2 b^6 c^14 k-1440 b^8 c^14 k+1200 a^6 c^16 k+1360 a^4 b^2 c^16 k+1360 a^2 b^4 c^16 k+1200 b^6 c^16 k-560 a^4 c^18 k-800 a^2 b^2 c^18 k-560 b^4 c^18 k+144 a^2 c^20 k+144 b^2 c^20 k-16 c^22 k+224 a^20 k^2-1984 a^18 b^2 k^2+8032 a^16 b^4 k^2-19712 a^14 b^6 k^2+32704 a^12 b^8 k^2-38528 a^10 b^10 k^2+32704 a^8 b^12 k^2-19712 a^6 b^14 k^2+8032 a^4 b^16 k^2-1984 a^2 b^18 k^2+224 b^20 k^2-1984 a^18 c^2 k^2+11840 a^16 b^2 c^2 k^2-29440 a^14 b^4 c^2 k^2+37120 a^12 b^6 c^2 k^2-17536 a^10 b^8 c^2 k^2-17536 a^8 b^10 c^2 k^2+37120 a^6 b^12 c^2 k^2-29440 a^4 b^14 c^2 k^2+11840 a^2 b^16 c^2 k^2-1984 b^18 c^2 k^2+8032 a^16 c^4 k^2-29440 a^14 b^2 c^4 k^2+40576 a^12 b^4 c^4 k^2-25856 a^10 b^6 c^4 k^2+13376 a^8 b^8 c^4 k^2-25856 a^6 b^10 c^4 k^2+40576 a^4 b^12 c^4 k^2-29440 a^2 b^14 c^4 k^2+8032 b^16 c^4 k^2-19712 a^14 c^6 k^2+37120 a^12 b^2 c^6 k^2-25856 a^10 b^4 c^6 k^2+8448 a^8 b^6 c^6 k^2+8448 a^6 b^8 c^6 k^2-25856 a^4 b^10 c^6 k^2+37120 a^2 b^12 c^6 k^2-19712 b^14 c^6 k^2+32704 a^12 c^8 k^2-17536 a^10 b^2 c^8 k^2+13376 a^8 b^4 c^8 k^2+8448 a^6 b^6 c^8 k^2+13376 a^4 b^8 c^8 k^2-17536 a^2 b^10 c^8 k^2+32704 b^12 c^8 k^2-38528 a^10 c^10 k^2-17536 a^8 b^2 c^10 k^2-25856 a^6 b^4 c^10 k^2-25856 a^4 b^6 c^10 k^2-17536 a^2 b^8 c^10 k^2-38528 b^10 c^10 k^2+32704 a^8 c^12 k^2+37120 a^6 b^2 c^12 k^2+40576 a^4 b^4 c^12 k^2+37120 a^2 b^6 c^12 k^2+32704 b^8 c^12 k^2-19712 a^6 c^14 k^2-29440 a^4 b^2 c^14 k^2-29440 a^2 b^4 c^14 k^2-19712 b^6 c^14 k^2+8032 a^4 c^16 k^2+11840 a^2 b^2 c^16 k^2+8032 b^4 c^16 k^2-1984 a^2 c^18 k^2-1984 b^2 c^18 k^2+224 c^20 k^2-4352 a^18 k^3+30464 a^16 b^2 k^3-87040 a^14 b^4 k^3+121856 a^12 b^6 k^3-60928 a^10 b^8 k^3-60928 a^8 b^10 k^3+121856 a^6 b^12 k^3-87040 a^4 b^14 k^3+30464 a^2 b^16 k^3-4352 b^18 k^3+30464 a^16 c^2 k^3-96256 a^14 b^2 c^2 k^3-31744 a^12 b^4 c^2 k^3+505856 a^10 b^6 c^2 k^3-816640 a^8 b^8 c^2 k^3+505856 a^6 b^10 c^2 k^3-31744 a^4 b^12 c^2 k^3-96256 a^2 b^14 c^2 k^3+30464 b^16 c^2 k^3-87040 a^14 c^4 k^3-31744 a^12 b^2 c^4 k^3+617472 a^10 b^4 c^4 k^3-498688 a^8 b^6 c^4 k^3-498688 a^6 b^8 c^4 k^3+617472 a^4 b^10 c^4 k^3-31744 a^2 b^12 c^4 k^3-87040 b^14 c^4 k^3+121856 a^12 c^6 k^3+505856 a^10 b^2 c^6 k^3-498688 a^8 b^4 c^6 k^3-258048 a^6 b^6 c^6 k^3-498688 a^4 b^8 c^6 k^3+505856 a^2 b^10 c^6 k^3+121856 b^12 c^6 k^3-60928 a^10 c^8 k^3-816640 a^8 b^2 c^8 k^3-498688 a^6 b^4 c^8 k^3-498688 a^4 b^6 c^8 k^3-816640 a^2 b^8 c^8 k^3-60928 b^10 c^8 k^3-60928 a^8 c^10 k^3+505856 a^6 b^2 c^10 k^3+617472 a^4 b^4 c^10 k^3+505856 a^2 b^6 c^10 k^3-60928 b^8 c^10 k^3+121856 a^6 c^12 k^3-31744 a^4 b^2 c^12 k^3-31744 a^2 b^4 c^12 k^3+121856 b^6 c^12 k^3-87040 a^4 c^14 k^3-96256 a^2 b^2 c^14 k^3-87040 b^4 c^14 k^3+30464 a^2 c^16 k^3+30464 b^2 c^16 k^3-4352 c^18 k^3+14592 a^16 k^4+47104 a^14 b^2 k^4-574464 a^12 b^4 k^4+1640448 a^10 b^6 k^4-2255360 a^8 b^8 k^4+1640448 a^6 b^10 k^4-574464 a^4 b^12 k^4+47104 a^2 b^14 k^4+14592 b^16 k^4+47104 a^14 c^2 k^4-1284096 a^12 b^2 c^2 k^4+3569664 a^10 b^4 c^2 k^4-2332672 a^8 b^6 c^2 k^4-2332672 a^6 b^8 c^2 k^4+3569664 a^4 b^10 c^2 k^4-1284096 a^2 b^12 c^2 k^4+47104 b^14 c^2 k^4-574464 a^12 c^4 k^4+3569664 a^10 b^2 c^4 k^4-195584 a^8 b^4 c^4 k^4-5599232 a^6 b^6 c^4 k^4-195584 a^4 b^8 c^4 k^4+3569664 a^2 b^10 c^4 k^4-574464 b^12 c^4 k^4+1640448 a^10 c^6 k^4-2332672 a^8 b^2 c^6 k^4-5599232 a^6 b^4 c^6 k^4-5599232 a^4 b^6 c^6 k^4-2332672 a^2 b^8 c^6 k^4+1640448 b^10 c^6 k^4-2255360 a^8 c^8 k^4-2332672 a^6 b^2 c^8 k^4-195584 a^4 b^4 c^8 k^4-2332672 a^2 b^6 c^8 k^4-2255360 b^8 c^8 k^4+1640448 a^6 c^10 k^4+3569664 a^4 b^2 c^10 k^4+3569664 a^2 b^4 c^10 k^4+1640448 b^6 c^10 k^4-574464 a^4 c^12 k^4-1284096 a^2 b^2 c^12 k^4-574464 b^4 c^12 k^4+47104 a^2 c^14 k^4+47104 b^2 c^14 k^4+14592 c^16 k^4+147456 a^14 k^5-1261568 a^12 b^2 k^5+2899968 a^10 b^4 k^5-1785856 a^8 b^6 k^5-1785856 a^6 b^8 k^5+2899968 a^4 b^10 k^5-1261568 a^2 b^12 k^5+147456 b^14 k^5-1261568 a^12 c^2 k^5+1802240 a^10 b^2 c^2 k^5+12533760 a^8 b^4 c^2 k^5-26148864 a^6 b^6 c^2 k^5+12533760 a^4 b^8 c^2 k^5+1802240 a^2 b^10 c^2 k^5-1261568 b^12 c^2 k^5+2899968 a^10 c^4 k^5+12533760 a^8 b^2 c^4 k^5-23822336 a^6 b^4 c^4 k^5-23822336 a^4 b^6 c^4 k^5+12533760 a^2 b^8 c^4 k^5+2899968 b^10 c^4 k^5-1785856 a^8 c^6 k^5-26148864 a^6 b^2 c^6 k^5-23822336 a^4 b^4 c^6 k^5-26148864 a^2 b^6 c^6 k^5-1785856 b^8 c^6 k^5-1785856 a^6 c^8 k^5+12533760 a^4 b^2 c^8 k^5+12533760 a^2 b^4 c^8 k^5-1785856 b^6 c^8 k^5+2899968 a^4 c^10 k^5+1802240 a^2 b^2 c^10 k^5+2899968 b^4 c^10 k^5-1261568 a^2 c^12 k^5-1261568 b^2 c^12 k^5+147456 c^14 k^5-3145728 a^10 b^2 k^6+12582912 a^8 b^4 k^6-18874368 a^6 b^6 k^6+12582912 a^4 b^8 k^6-3145728 a^2 b^10 k^6-3145728 a^10 c^2 k^6+34603008 a^8 b^2 c^2 k^6-31457280 a^6 b^4 c^2 k^6-31457280 a^4 b^6 c^2 k^6+34603008 a^2 b^8 c^2 k^6-3145728 b^10 c^2 k^6+12582912 a^8 c^4 k^6-31457280 a^6 b^2 c^4 k^6-96468992 a^4 b^4 c^4 k^6-31457280 a^2 b^6 c^4 k^6+12582912 b^8 c^4 k^6-18874368 a^6 c^6 k^6-31457280 a^4 b^2 c^6 k^6-31457280 a^2 b^4 c^6 k^6-18874368 b^6 c^6 k^6+12582912 a^4 c^8 k^6+34603008 a^2 b^2 c^8 k^6+12582912 b^4 c^8 k^6-3145728 a^2 c^10 k^6-3145728 b^2 c^10 k^6+67108864 a^6 b^2 c^2 k^7-134217728 a^4 b^4 c^2 k^7+67108864 a^2 b^6 c^2 k^7-134217728 a^4 b^2 c^4 k^7-134217728 a^2 b^4 c^4 k^7+67108864 a^2 b^2 c^6 k^7 = 0

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hujunhua

何校长提出的三角形的Yff点因为是一个3次方程的根，我都觉得不是什么好"点"，没想到这里是7次方程，就更不是什么好心了！

还是内心好哇，对应着垂心！

hujunhua

发现存在一个简明的转换公式.

如果`O`对`△ABC`的重心坐标是`(α:β:γ)`, `(α+β+γ=1)`,
那么O对其基于△ABC的Ceva三角形△DEF的面积坐标是\[
\left(\frac{β+γ}2:\frac{γ+α}2:\frac{α+β}2\right)=\left(\frac{1-α}2:\frac{1-β}2:\frac{1-γ}2\right)
\]反过来，如果O对Ceva三角形△DEF的重心坐标是`(α:β:γ)`, `(α+β+γ=1)`，
那么O对△ABC的面积坐标是\[
\left((β+γ-α) : (γ+α-β) : (α+β-γ)\right)=\left((1-2α) : (1-2β) : (1-2γ)\right)
\]