# Find the length of BC.

A rectangle HFMO has sides HO=7 and OM=12. A triangle ABC has H as its orthocentre, O as its circumcentre, M as the midpoint of BC, and F as the foot of the altitude from A. Find the length of BC.

Pak Wai sir,

You mean orthocentre, centriod and circumcentre should be collinear ? Why ?

Therefore, X lies on the height AF. ⋯⋯(agree)

Similarly prove for other sides with their medians, we have X lies on three heights.⋯⋯(?, 3?)

i.e. X is the orthocenter H and O, G, H are collinear. ⋯⋯ (why X is H?)

Anyway, I have checked that OGH are collinear due to the Euler line. Thx.

Pak Wai sir 的解法無誤，只是 DSE 冇教，所以答了也不可能拿滿分。

### 8 個解答

• 最愛解答

HO // FM (rectangle properties)

Joint AM that intersects HO at the centroid G.

so, AM = 3GM

hence AF = 3FH = 36 (ΔAOH ~ ΔAFM since equilangular)

AH = AF - HF = 36 - 12 = 24

Joint OA and OC where OA = OB = OC = radius R

OA^2 = OH^2 + HA^2

OA = (7^2 + 24^2)^(1/2) = 25

OC = AO = 25

CM^2 = OC^2 - OM^2

CM = (25^2 - 12^2)^(1/2) = 481^(1/2)

BC = 2CM = 2[481^(1/2)]

2015-01-18 03:14:40 補充：

It should be ΔAHG ~ ΔAFM since equilangular

∠GAH = ∠MAF (common ∠)

∠AGH = ∠AMF (corr. ∠, GH//FM)

∠AHG = ∠AFM (corr. ∠, GH//FM)

2015-01-18 12:18:50 補充：

In this question, let G be the centroid.

Extend OG until a certain point, saids X, such that OG = 2GX.

GX = 2GO (we set this)

AG = 2MG since G is the centroid

∠OGM = ∠XGA (vert. opp. ∠s)

i.e. ΔOGM ~ ΔXGA (2 sides in ratio, int. ∠s)

(to be cont.)

2015-01-18 12:24:58 補充：

(It should be OG = (1/2)GX, typing error.)

We have

∠OMG = ∠OAX => AX // OM

But OM ⊥ BC => AX ⊥ BC

Therefore, X lies on the height AF.

Similarly prove for other sides with their medians, we have X lies on three heights.

i.e. X is the orthocenter H and O, G, H are collinear.

Note that 2OG = GH

2015-01-18 12:27:14 補充：

In fact, the line passes through the centroid, the circumcenter and the orthocenter of a triangle is called its Euler line.

2015-01-18 12:32:53 補充：

Sorry for my typing errors.

The last few lines should be

∠OMG = ∠XAG => AX // OM

But OM ⊥ BC => AX ⊥ BC

(again, sorry for my typing errors =])

2015-01-18 12:46:03 補充：

The graph could be drawn like:

http://i1379.photobucket.com/albums/ah127/jackiekw...

2015-01-18 17:13:43 補充：

題目給了HO是歐拉線……就是不知道用不用證明它 0 0

2015-01-18 19:20:20 補充：

題目冇明示，不過有外心同垂心，好自然就諗咗去Euler line果邊 XD

2015-01-19 11:13:50 補充：

Similarly prove for other sides with their medians, we have X lies on three heights.⋯⋯(?, 3?)

It is to prove that the three heights of the triangle pass through X.

There are 3 heights, one as AH with the vertex A and the foot H of the height AH.

2015-01-19 11:21:06 補充：

Repeating this process for the same point X with different vertices (B, C) and their feet of the altitude, and you could prove that all the heights pass through X. Hence X is the orthocenter, where we usually called H.

2015-01-19 11:24:09 補充：

AF, not AH = =

2015-01-19 11:30:30 補充：

我經常打錯字…… QQ

Co-geom簡單易懂，可是沒有想到這邊。

對了，你們是用WORD打的嗎？

2015-01-19 12:10:48 補充：

我是沒有用過這些數學軟件……QQ

資料來源： Myself, Myself, Myself, Myself., Myself, Myself, Myself, Myself, Myself, Myself
• 登入以回覆解答
• Co-geom呢類係唔錯

http://postimg.org/image/3ztr39vs1/

2015-01-19 11:44:17 補充：

Re：少年時

謝謝 =)

let B嘅y-coordinate係y的確會快好多 XD

Re：Pak Wai

我用慣哥個叫 MathType

感覺上覺得幾 OK

• 登入以回覆解答
• 貓sir, 題目似乎是講 Geometry, 但解決方法不一定要用 Geometry,

例如用 Co-geom 之類.

2015-01-19 11:04:22 補充：

一點即明，管理員的DSE成績一定好好了。不過方法可改進：

Let O be the origin, so, H=(0,7), F=(12,7), M=(12,0)

Let A=(-x,7), B=(12,y), C＝(12,-y)

As OA=OB, so,

x²＋7²＝12²＋y²

==> x²＋49＝144＋y² ⋯⋯ (i)

And BH⊥AC, so,

(y－7)/12 * (7＋y)/(-x－12) = -1

==> y²－49＝12x＋144 ⋯⋯ (ii)

2015-01-19 11:05:25 補充：

(i)＋(ii), get,

x²＝12x＋288

==> (x－24)(x＋12)＝0

==> x＝24 or -12 (rej)

Sub into (i), get y = √481

∴ BC＝2x＝2√481

2015-01-19 11:07:35 補充：

如果懂得 Euler line, Pak Wai 的解答是最方便快捷的.

• 登入以回覆解答
• 其實我不是太熟悉 geometry (幾何學) 的課題。

或者看看 雨後 前輩 是否有空。

• 登入以回覆解答
• 你覺得這個解答怎樣？你可以登入投選解答。
• 貓貓會答這個嗎?我也不太會...

• 登入以回覆解答
• 開估啦!!!!!!!!

• 登入以回覆解答
• 左圖畫好些是做到的。

2015-01-17 08:32:21 補充：

但依家係想計 BC 有幾長，唔係想量出 BC 有幾長。答案有開方符號的。

2015-01-18 18:27:29 補充：

其實題目冇提到乜野歐拉線, 而且呢題數我喺DSE啲mock卷揾出來.

照我所知佢地(中六)應該都唔知什麼是Euler line, 無論如何, 你的答案是正確的.

睇吓有冇其他高手可以唔使用Euler line來做吧.

• 登入以回覆解答
• 左圖H不是orthocentre,但HFMO是長方形

右圖H是orthocentre,但HFMO不是長方形

△ABC要在圓周上,而O不能在△ABC之外

B,C,M,O已經定位,只能透過A點來定位F&H

問題有否出錯?

• 登入以回覆解答