Bước 1: Chứng minh △MNK=△ENKtriangle cap M cap N cap K equals triangle cap E cap N cap K
△𝑀𝑁𝐾=△𝐸𝑁𝐾
 
Xét △MNKtriangle cap M cap N cap K
△𝑀𝑁𝐾
và △ENKtriangle cap E cap N cap K
△𝐸𝑁𝐾
có: 
MN=NEcap M cap N equals cap N cap E
𝑀𝑁=𝑁𝐸
(giả thiết).
MNK̂=ENK̂modified cap M cap N cap K with hat above equals modified cap E cap N cap K with hat above
𝑀𝑁𝐾=𝐸𝑁𝐾
(do NKcap N cap K
𝑁𝐾
là tia phân giác của MNP̂modified cap M cap N cap P with hat above
𝑀𝑁𝑃
).
NKcap N cap K
𝑁𝐾
là cạnh chung. 

Vậy △MNK=△ENKtriangle cap M cap N cap K equals triangle cap E cap N cap K
△𝑀𝑁𝐾=△𝐸𝑁𝐾
theo trường hợp cạnh - góc - cạnh (c.g.c). 

Bước 2: Chứng minh IE⟂NPcap I cap E ⟂ cap N cap P
𝐼𝐸⟂𝑁𝑃
 
Vì Icap I
𝐼
là giao điểm của NKcap N cap K
𝑁𝐾
và MPcap M cap P
𝑀𝑃
, mà NKcap N cap K
𝑁𝐾
là tia phân giác của MNP̂modified cap M cap N cap P with hat above
𝑀𝑁𝑃
, nên Icap I
𝐼
cũng nằm trên tia phân giác của góc Ncap N
𝑁
.
Xét △MNItriangle cap M cap N cap I
△𝑀𝑁𝐼
và △ENItriangle cap E cap N cap I
△𝐸𝑁𝐼
có: 
MN=NEcap M cap N equals cap N cap E
𝑀𝑁=𝑁𝐸
(giả thiết).
MNÎ=ENÎmodified cap M cap N cap I with hat above equals modified cap E cap N cap I with hat above
𝑀𝑁𝐼=𝐸𝑁𝐼
(do NIcap N cap I
𝑁𝐼
là phân giác).
NIcap N cap I
𝑁𝐼
là cạnh chung. 

⇒△MNI=△ENIimplies triangle cap M cap N cap I equals triangle cap E cap N cap I
⇒△𝑀𝑁𝐼=△𝐸𝑁𝐼
(c.g.c).
⇒NMÎ=NEÎimplies modified cap N cap M cap I with hat above equals modified cap N cap E cap I with hat above
⇒𝑁𝑀𝐼=𝑁𝐸𝐼
(hai góc tương ứng).
Mà NMÎ=90∘modified cap N cap M cap I with hat above equals 90 raised to the composed with power
𝑁𝑀𝐼=90∘
(do △MNPtriangle cap M cap N cap P
△𝑀𝑁𝑃
vuông tại Mcap M
𝑀
), nên NEÎ=90∘modified cap N cap E cap I with hat above equals 90 raised to the composed with power
𝑁𝐸𝐼=90∘
.
Vậy IE⟂NPcap I cap E ⟂ cap N cap P
𝐼𝐸⟂𝑁𝑃
. 

Bước 3: Chứng minh MNÎ=QEP̂modified cap M cap N cap I with hat above equals modified cap Q cap E cap P with hat above
𝑀𝑁𝐼=𝑄𝐸𝑃
 
Vì EF∥MPcap E cap F is parallel to cap M cap P
𝐸𝐹∥𝑀𝑃
và NIcap N cap I
𝑁𝐼
cắt chúng, nên EFÎ=MIN̂modified cap E cap F cap I with hat above equals modified cap M cap I cap N with hat above
𝐸𝐹𝐼=𝑀𝐼𝑁
(hai góc so le trong).
Mặt khác, từ △MNI=△ENItriangle cap M cap N cap I equals triangle cap E cap N cap I
△𝑀𝑁𝐼=△𝐸𝑁𝐼
, ta có MIN̂=EIN̂modified cap M cap I cap N with hat above equals modified cap E cap I cap N with hat above
𝑀𝐼𝑁=𝐸𝐼𝑁
.
Suy ra EFÎ=EIF̂modified cap E cap F cap I with hat above equals modified cap E cap I cap F with hat above
𝐸𝐹𝐼=𝐸𝐼𝐹
(vì F,Icap F comma cap I
𝐹,𝐼
cùng nằm trên đường thẳng NIcap N cap I
𝑁𝐼
), dẫn đến △IEFtriangle cap I cap E cap F
△𝐼𝐸𝐹
cân tại Ecap E
𝐸
. Do đó EF=EIcap E cap F equals cap E cap I
𝐸𝐹=𝐸𝐼
.
Theo giả thiết IQ=FEcap I cap Q equals cap F cap E
𝐼𝑄=𝐹𝐸
, nên IQ=EIcap I cap Q equals cap E cap I
𝐼𝑄=𝐸𝐼
. Tam giác IEQcap I cap E cap Q
𝐼𝐸𝑄
cân tại Icap I
𝐼
.
Đặt MNÎ=α⇒ENÎ=αmodified cap M cap N cap I with hat above equals alpha implies modified cap E cap N cap I with hat above equals alpha
𝑀𝑁𝐼=𝛼⇒𝐸𝑁𝐼=𝛼
. Trong tam giác vuông MNIcap M cap N cap I
𝑀𝑁𝐼
, ta có MIN̂=90∘−αmodified cap M cap I cap N with hat above equals 90 raised to the composed with power minus alpha
𝑀𝐼𝑁=90∘−𝛼
.
Vì M,I,Pcap M comma cap I comma cap P
𝑀,𝐼,𝑃
thẳng hàng, góc bẹt MIP̂=180∘modified cap M cap I cap P with hat above equals 180 raised to the composed with power
𝑀𝐼𝑃=180∘
. Ta có:
EIP̂=180∘−(MIN̂+EIN̂)=180∘−2(90∘−α)=2αmodified cap E cap I cap P with hat above equals 180 raised to the composed with power minus open paren modified cap M cap I cap N with hat above plus modified cap E cap I cap N with hat above close paren equals 180 raised to the composed with power minus 2 open paren 90 raised to the composed with power minus alpha close paren equals 2 alpha
𝐸𝐼𝑃=180∘−(𝑀𝐼𝑁+𝐸𝐼𝑁)=180∘−2(90∘−𝛼)=2𝛼

Trong tam giác cân IEQcap I cap E cap Q
𝐼𝐸𝑄
( IE=IQcap I cap E equals cap I cap Q
𝐼𝐸=𝐼𝑄
):
IEQ̂=180∘−EIP̂2=180∘−2α2=90∘−αmodified cap I cap E cap Q with hat above equals the fraction with numerator 180 raised to the composed with power minus modified cap E cap I cap P with hat above and denominator 2 end-fraction equals the fraction with numerator 180 raised to the composed with power minus 2 alpha and denominator 2 end-fraction equals 90 raised to the composed with power minus alpha
𝐼𝐸𝑄=180∘−𝐸𝐼𝑃2=180∘−2𝛼2=90∘−𝛼

Vì IE⟂NPcap I cap E ⟂ cap N cap P
𝐼𝐸⟂𝑁𝑃
, nên IEP̂=90∘modified cap I cap E cap P with hat above equals 90 raised to the composed with power
𝐼𝐸𝑃=90∘
. Ta có:
QEP̂=IEP̂−IEQ̂=90∘−(90∘−α)=αmodified cap Q cap E cap P with hat above equals modified cap I cap E cap P with hat above minus modified cap I cap E cap Q with hat above equals 90 raised to the composed with power minus open paren 90 raised to the composed with power minus alpha close paren equals alpha
𝑄𝐸𝑃=𝐼𝐸𝑃−𝐼𝐸𝑄=90∘−(90∘−𝛼)=𝛼

Vậy MNÎ=QEP̂modified cap M cap N cap I with hat above equals modified cap Q cap E cap P with hat above
𝑀𝑁𝐼=𝑄𝐸𝑃
(cùng bằng αalpha
𝛼
). 

Đáp án: 
a) △MNK=△ENKtriangle cap M cap N cap K equals triangle cap E cap N cap K
△𝑀𝑁𝐾=△𝐸𝑁𝐾
(c.g.c).
b) IE⟂NPcap I cap E ⟂ cap N cap P
𝐼𝐸⟂𝑁𝑃
do NEÎ=NMÎ=90∘modified cap N cap E cap I with hat above equals modified cap N cap M cap I with hat above equals 90 raised to the composed with power
𝑁𝐸𝐼=𝑁𝑀𝐼=90∘
.
c) MNÎ=QEP̂modified cap M cap N cap I with hat above equals modified cap Q cap E cap P with hat above
𝑀𝑁𝐼=𝑄𝐸𝑃
.