數學問題一個(自問係數學人請進)

I cannot prove K exists, but in fact, I have found a way to draw ABCDE in which K does NOT exist:

The question statement does not state that all internal angles are smaller than 180deg. Imagine ABDE forms a square while BCD is a reflex angle (like an M with a closed bottom)

AD, being the diagonal of the square ABDE, is still the longest diagonal in ABCDE.

In this case, no K can possible be located on AD within the bound of A and D which satisfies:
I) BKA less than 90deg
II) BKC less than 90deg
III) CKD less than 90deg
at the same point.

Points to satisfy I) are located on AD between the mid-point and point D
Points to satify III), however, are located between the right-angle projection of point C on AD and point A.

The ranges of points to satisfy I) and III) do not overlap.

The statement can therefore never be proven generally true if evidence to the contrary can be found. Not unless the statement itself is inaccurate or incomplete, e.g. there are no reflex angles.

2008-04-05 06:01:11 補充：
Can't understand why there is still universal proof if a contradictive case can be found...

答前想問一問用中文+英文答, 可以嗎?
我prove了, 製圖中

2008-04-01 20:12:41 補充：
圖:
http://photo.xanga.com/ho_Paco/28666181908558/photo.html

∠A+∠B+∠C+∠D+∠E = (5-2)*180 (∠sum of polygon)
∠B=108

∵AB=BC(given)
∴∠cab = ∠bca (base ∠s, isos. △)
∠B +∠cab+∠bca=180 (∠sum of △)

∠cab=∠bca=36

∠dca+∠bca=108
∠dca =72

AD=AC (given)
∠cda= ∠dca=72

2008-04-01 20:13:00 補充：
Line CK(blue line) ┴AE and ABCDE is a regular polygon
∠dck=∠bck =108 / 2
∠dck =54

∠dck+∠cda+∠ckd=180(∠sum of △)
∠ckd=180-72-54
∠ckd=54(∠CKD＜90度)

∵BE is parallel to CD
∴∠apk=∠acd (corr. ∠s, BE parallel CD)
∠apk=72
∠apk+∠cpk=180(adj. ∠s on st. line)
∠cpk=108

2008-04-01 20:13:04 補充：
∠dck+∠bca+∠ack=108
∠ack=108-54-36
∠ack=18

∠ack+∠cpk+∠bkc=180(∠sum of △)
∠bkc=180-18-108
∠bkc=54(∠BKC＜90度)

Lastly,
∠bkc+∠ckd+∠bka=180(adj. ∠s on st. line)
∠bka=180-54-54
∠bka=72(∠BKA＜90度)

希望不要刪除問題, 浪費答題者心血

2008-04-04 00:38:52 補充：
五邊相等和五角相等的五邊形是正五邊形, 不是嗎?
我沒說只有五邊相等
my explaination有計五角相等

另外, 還請指教指教

因為無數字好難用實數求證
,但五條邊相等, 再係AD 畫一條線, 在AD 的中間點為K
在KB 同KC 分別畫一條線, 變成三個一樣的三角形
, 再細睇, AED 係一個(等腰三角形), ABCD係一個梯形
咁再係AC 上面畫多一條線, AD 同AC一樣長度(一個等腰三角形)
咁就睇到AD / AC係最長的一條線
我覺得而已

因為A和D是距離最遠的.是不是?.......

AD中間-.-|||.........................