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.

更新:

Pak Wai sir,

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

更新 2:

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.

更新 3:

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

反而 管理員 及 少年時 的可得滿分的。

無論如何,多謝三位的解說。

新年進步!

8 個解答

評分
  • 6 年前
    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?

    (比如這張:http://postimg.org/image/3ztr39vs1/)

    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
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  • 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

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  • 6 年前

    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.

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  • 6 年前

    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。

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  • Tony
    Lv 4
    6 年前

    貓貓會答這個嗎?我也不太會...

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  • 6 年前

    開估啦!!!!!!!!

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  • 6 年前

    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.

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  • 6 年前

    http://oi57.tinypic.com/1hebsl.jpg (比例有出入)

    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?

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