Giả sử cạnh của tam giác đều ABCcap A cap B cap C
???
là 2a2 a
2?
. Khi đó, đường cao AH=a3cap A cap H equals a the square root of 3 end-root
??=?3√
và HC=acap H cap C equals a
??=?
.
Theo giả thiết, DH=AH=a3cap D cap H equals cap A cap H equals a the square root of 3 end-root
??=??=?3√
.
Xét điểm E′cap E prime
?′
trên tia ACcap A cap C
??
sao cho CE′=HC=acap C cap E prime equals cap H cap C equals a
??′=??=?
. Ta sẽ chứng minh E′cap E prime
?′
trùng với Ecap E
?
. 
Trong ΔHCE′cap delta cap H cap C cap E prime
Δ???′
, ta có HC=CE′=acap H cap C equals cap C cap E prime equals a
??=??′=?
và ∠HCE′=180∘−∠ACB=120∘angle cap H cap C cap E prime equals 180 raised to the composed with power minus angle cap A cap C cap B equals 120 raised to the composed with power
∠???′=180∘−∠???=120∘
(do A,C,E′cap A comma cap C comma cap E prime
?,?,?′
thẳng hàng).
ΔHCE′cap delta cap H cap C cap E prime
Δ???′
cân tại C⇒∠CHE′=∠CE′H=180∘−120∘2=30∘cap C implies angle cap C cap H cap E prime equals angle cap C cap E prime cap H equals the fraction with numerator 180 raised to the composed with power minus 120 raised to the composed with power and denominator 2 end-fraction equals 30 raised to the composed with power
?⇒∠???′=∠??′?=180∘−120∘2=30∘
.
Áp dụng định lý hàm số cos trong ΔHCE′cap delta cap H cap C cap E prime
Δ???′
:
HE′2=HC2+CE′2−2HC⋅CE′⋅cos(120∘)=a2+a2−2a2(−12)=3a2cap H cap E prime squared equals cap H cap C squared plus cap C cap E prime squared minus 2 cap H cap C center dot cap C cap E prime center dot cosine open paren 120 raised to the composed with power close paren equals a squared plus a squared minus 2 a squared open paren negative one-half close paren equals 3 a squared
??′2=??2+??′2−2??⋅??′⋅cos(120∘)=?2+?2−2?2(−12)=3?2

Suy ra HE′=a3cap H cap E prime equals a the square root of 3 end-root
??′=?3√
. Vì DH=a3cap D cap H equals a the square root of 3 end-root
??=?3√
nên HE′=DHcap H cap E prime equals cap D cap H
??′=??
.
Xét ΔDHE′cap delta cap D cap H cap E prime
Δ???′
, ta có HD=HE′=a3cap H cap D equals cap H cap E prime equals a the square root of 3 end-root
??=??′=?3√
nên ΔDHE′cap delta cap D cap H cap E prime
Δ???′
cân tại Hcap H
?
.
Góc ∠DHE′=180∘−∠CHE′=180∘−30∘=150∘angle cap D cap H cap E prime equals 180 raised to the composed with power minus angle cap C cap H cap E prime equals 180 raised to the composed with power minus 30 raised to the composed with power equals 150 raised to the composed with power
∠???′=180∘−∠???′=180∘−30∘=150∘
.
⇒∠HDE′=180∘−150∘2=15∘implies angle cap H cap D cap E prime equals the fraction with numerator 180 raised to the composed with power minus 150 raised to the composed with power and denominator 2 end-fraction equals 15 raised to the composed with power
⇒∠???′=180∘−150∘2=15∘
.
Vì ∠HDE′=15∘angle cap H cap D cap E prime equals 15 raised to the composed with power
∠???′=15∘
trùng với giả thiết ∠BDx=15∘angle cap B cap D x equals 15 raised to the composed with power
∠???=15∘
và Ecap E
?
thuộc tia ACcap A cap C
??
nên E≡E′cap E triple bar cap E prime
?≡?′
.
Vậy EH=DHcap E cap H equals cap D cap H
??=??
và HC=ECcap H cap C equals cap E cap C
??=??
. 

Bước 2: Chứng minh EH⟂ABcap E cap H ⟂ cap A cap B
??⟂??
 
Gọi Kcap K
?
là giao điểm của đường thẳng EHcap E cap H
??
và ABcap A cap B
??
. 
Từ kết quả câu (a), ta có ∠CHE=30∘angle cap C cap H cap E equals 30 raised to the composed with power
∠???=30∘
.
Vì Hcap H
?
nằm giữa Bcap B
?
và Ccap C
?
, tia HEcap H cap E
??
nằm ở nửa mặt phẳng dưới BCcap B cap C
??
, nên góc đối đỉnh của ∠CHEangle cap C cap H cap E
∠???
là góc tạo bởi tia đối của HEcap H cap E
??
và tia HBcap H cap B
??
. Gọi góc này là ∠BHK=30∘angle cap B cap H cap K equals 30 raised to the composed with power
∠???=30∘
.
Trong ΔABCcap delta cap A cap B cap C
Δ???
đều, ∠ABC=60∘angle cap A cap B cap C equals 60 raised to the composed with power
∠???=60∘
hay ∠KBH=60∘angle cap K cap B cap H equals 60 raised to the composed with power
∠???=60∘
.
Xét tam giác BKHcap B cap K cap H
???
, ta có:
∠BKH=180∘−(∠KBH+∠BHK)=180∘−(60∘+30∘)=90∘angle cap B cap K cap H equals 180 raised to the composed with power minus open paren angle cap K cap B cap H plus angle cap B cap H cap K close paren equals 180 raised to the composed with power minus open paren 60 raised to the composed with power plus 30 raised to the composed with power close paren equals 90 raised to the composed with power
∠???=180∘−(∠???+∠???)=180∘−(60∘+30∘)=90∘

Vậy EH⟂ABcap E cap H ⟂ cap A cap B
??⟂??
tại Kcap K
?
. 

Đáp án: 
a. EH=DHcap E cap H equals cap D cap H
??=??
(cùng bằng AHcap A cap H
??
)
b. HC=ECcap H cap C equals cap E cap C
??=??

c. EH⟂ABcap E cap H ⟂ cap A cap B
??⟂??
tại giao điểm của chúng. 

 
 
 
 
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